Area of a Triangle
1996年美国数学竞赛-AMC_12
19961The addition below is incorrect.What is the largest digit that can be changed to make the addition correct?641852+9732456(A)4(B)5(C)6(D)7(E)8Each day Walter gets $3for doing his chores or $5for doing them exceptionally well.After 10days of doing his chores daily,Walter has received a total of $36.On how many days did Walter do them exceptionally well?A.3B.4C.5D.6E.7(3!)!3!=(A)1(B)2(C)6(D)40(E)120Six numbers from a list of nine integers are 7,8,3,5,and 9.The largest possible value of the median of all nine numbers in this list is(A)5(B)(C)7(D)8(E)Given that 0<a <b <c <d ,which of the following is the largest?A.a +b c +d B.a +d b +c C.b +c a +d D.b +d a +c E.c +d a +bIf f (x )=x (x +1)(x +2)(x +3)then f (0)+f (−1)+f (−2)+f (−3)=A.−8/9B.0C.8/9D.1E.10/9A father takes his twins and a younger child out to dinner on the twins’birthday.The restaurant charges $4.95for the father and $0.45for each year of a child’s age,where age is defined as the age at the most recent birthday.If the bill is $9.45,which of the following could be the age of the youngest child?A.1B.2C.3D.4E.5If 3=k ·2r and 15=k ·4r ,then r =(A)−log 25(B)log 52(C)log 105(D)log 25(E)52Triangle P AB and square ABCD are in perpendicular planes.Given that P A =3,P B =4,and AB =5,what is P D ?A.5B.√34C.√41D.2√13E.8This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/1996How many line segments have both their endpoints located at the vertices of a given cube?(A)12(B)15(C)24(D)28(E)56Given a circle of raidus 2,there are many line segments of length 2that are tangent to the circle at their midpoints.Find the area of the region consisting of all such line segments.(A)π4(B)4−π(C)π2(D)π(E)2πA function f from the integers to the integers is defined as follows:f (x )= n +3if n is odd n/2if n is evenSuppose k is odd and f (f (f (k )))=27.What is the sum of the digits of k ?(A)3(B)6(C)9(D)12(E)15Sunny runs at a steady rate,and Moonbeam runs m times as fast,where m is a number greater than 1.If Moonbeam gives Sunny a head start of h meters,how many meters must Moonbeam run to overtake Sunny?(A)hm (B)h h +m (C)h m −1(D)hm m −1(E)h +m m −1Let E (n )denote the sum of the even digits of n .For example,E (5681)=6+8=14.Find E (1)+E (2)+E (3)+···+E (100).(A)200(B)360(C)400(D)900(E)2250Two opposite sides of a rectangle are each divided into n congruent segments,and the endpoints of one segment are joined to the center to form triangle A .The other sides are each divided into m congruent segments,and the endpoints of one of these segments are joined to the center to form triangle B .[See figure for n =5,m =7.]What is the ratio of the area of triangle A to the area of triangle B ?(A)1(B)m/n (C)n/m (D)2m/n (E)2n/mA fair standard six-sided dice is tossed three times.Given that the sum of the first two tosses equal the third,what is the probability that at least one ”2”is tossed?(A)16(B)91216(C)12(D)815(E)712In rectangle ABCD ,angle C is trisected by CF and CE ,where E is on AB ,F is on AD ,BE =6,and AF =2.Which of the following is closest to the area of the rectangle ABCD ?(A)110(B)120(C)130(D)140(E)150A circle of radius 2has center at (2,0).A circle of radius 1has center at (5,0).A line is tangent to the two circles at points in the first quadrant.Which of the following is closest to the y -intercept of the line?(A)√4(B)8/3(C)1+√(D)2√(E)3The midpoints of the sides of a regular hexagon ABCDEF are joined to form a smaller hexagon.What fraction of the area of ABCDEF is enclosed by the smaller hexagon?1996(A)12(B)√33(C)23(D)34(E)√32In the xy-plane,what is the length of the shortest path from (0,0)to (12,16)that does not go inside the circle (x −6)2+(y −8)2=25?(A)10√3(B)10√5(C)10√3+5π3(D)40√33(E)10+5πTriangles ABC and ABD are isosceles with AB =AC =BD ,and BD intersects AC at E .If BD is perpendicular to AC ,then ∠C +∠D is[img]/Forum/albump ic.php ?pic i d =537[/img ](A)115◦(B)120◦(C)130◦(D)135◦(E)not uniquely determinedFour distinct points,A ,B ,C ,and D ,are to be selected from 1996points evenly spaced around a circle.All quadruples are equally likely to be chosen.What is the probability that the chord AB intersects the chord CD?(A)14(B)13(C)12(D)23(E)34The sum of the lengths of the twelve edges of a rectangular box is 140,and the distance from one corner of the box to the farthest corner is 21.The total surface area of the box is(A)776(B)784(C)798(D)800(E)812The sequence 1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,...consists of 1s separated by blocks of 2s with n 2s in the nth block.The sum of the first 1234terms of this sequence is(A)1996(B)2419(C)2429(D)2439(E)2449Given that x 2+y 2=14x +6y +6,what is the largest possible value that 3x +4y can have?(A)72(B)73(C)74(D)75(E)76An urn contains marbles of four colors:red,white,blue,and green.When four marbles are drawn without replacement,the following events are equally likely:(a)the selection of four red marbles;(b)the selection of one white and three red marbles;(c)the selection of one white,one blue,and two red marbles;and (d)the selection of one marble of each color.What is the smallest number of marbles satisfying the given condition?(A)19(B)21(C)46(D)69Consider two solid spherical balls,one centered at (0,0,212)with radius 6,and the other centeredat (0,0,1)with radius 92.How many points (x,y,z )with only integer coordinates (lattice points)are there in the intersection of the balls?(A)7(B)9(C)11(D)13(E)15On a 4×4×3rectangular parallelepiped,vertices A ,B ,and C are adjacent to vertex D .The perpendicular distance from D to the plane containing A ,B ,and C is closest to(A)1.6(B)1.9(C)2.1(D)2.7(E)2.91996If n is a positive integer such that 2n has 28positive divisors and 3n has 30positive divisors,then how many positive divisors does 6n have?(A)32(B)34(C)35(D)36(E)38A hexagon inscribed in a circle has three consecutive sides each of length 3and three consecutive sides each of length 5.The chord of the circle that divides the hexagon into two trapezoids,one with three sides each of length 3and the other with three sides each of length 5,has length equal to m n ,where m and n are relatively prime positive integers.Find m +n .(A)309(B)349(C)369(D)389(E)409。
SAT考试数学练习题
SAT 考试数学练习题 2来源 : 91SAT 考试网时间 : 2009 年 08 月 20 日1. If f(x) = │(x² –50)│, what is the value of f( -5) ?A. 75B. 25C. 0D. -25E. -752. ( √2 - √3 )² =A. 5 - 2√6B. 5 - √6C. 1 - 2√6D. 1 - √2E. 13. 230 + 230 + 230 + 230 =A. 8120B. 830C. 232D. 230E. 2264. Amy has to visit towns B and C in any order. The roads connecting these towns with her home are shown on the diagram. How many different routes can she take starting from A and returning to A, going through both B and C (but not more than once through each) and not travelling any road twice on the same trip?A. 10B. 8C. 6D. 4E. 25. In the figure above AD = 4, AB = 3 and CD = 9. What is the area of triangle AEC ?A. 18B. 13.5C. 9D. 4.5E. 3答案：1.Correct Answer: BExplanation:If x = -5, then (x² – 50) = 25 – 50 = -25But the sign │x│ means the absolute value of x (the distance between the number and zero on the number line). Absolute values are always positive.│-25 │ = 252.Correct Answer: AExplanation:Expand as for (a + b)2.(√2 - √3)(√2 - √3) = 2 - 2(√2 + √3) + 3 = 5 - 2 √63.Correct Answer: CExplanation:All four terms are identical therefore we have 4 (230).But 4 = 22, and so we can write 22. 230Which is equivalent to 2324. Correct Answer: BExplanation:Amy can travel clockwise or anticlockwise on the diagram.Clockwise, she has no choice of route from A to B, a choice of one out of two routes from B to C, and a choice of one out of two routes from C back to A. This gives four possible routes. Similarly, anticlockwise she has four different routes.Total routes = 85.Correct Answer: DExplanation:If we take AE as the base of triangle AEC, then the height is CD.The height of the triangle is therefore, 9 (given).To find the base we need to see that triangles AEB and CDE are similar. The ratio AB: CD, is therefore equal to the ratio AE: ED. The given information shows that the ratio is 3:9, or 1:3. Now dividing AD (4) in this ratio gives us AE as 1.The area of AEC = ½ base x heigh t=1/2 x 9 = 4.5SAT 考试数学练习题 3来源 : 91SAT 考试网时间 : 2009 年 08 月 20 日1.Which of the following could be a value of x, in the diagram above?A. 10B. 20C. 40D. 50E.any of the above2.Helpers are needed to prepare for the fete.Each helper can make either 2 large cakes or 35small cakes per hour.The kitchen is available for 3hours and20 large cakes and 700 small cakes are needed. How many helpers are required?A. 10B. 15C. 20D. 25E. 303.Jo's collection contains US,Indian and British stamps.If the ratio of US to Indian stamps is 5to 2and the ratio of Indian to British stamps is 5to 1,what is the ratio of US to British stamps?A. 5 : 1B. 10 : 5C. 15 : 2D. 20 : 2E. 25 : 24.A 3by 4rectangle is inscribed in circle.What is the circumference of the circle?A.2.5πB.3πC.5πD.4πE. 10π5.Two sets of 4consecutive positive integers have exactly one integer in common. The sum of the integers in the set with greater numbers is how much greater than the sum of the integers in the other set?A. 4B.7C.8D. 12E. it cannot be determined from the information given.答案：1.Correct Answer: BExplanation:The marked angle, ABC must be more than 90 degrees because it is the external angle of triangle BDC, and must be equal to the sum of angles BDC (90) and DCB. Also ABC is not a straight line and must be less than 180. Therefore 90 < 5x < 180 The only value of x which satisfies this relation is 20.2.Correct Answer: AExplanation:20large cakes will require the equivalent of 10helpers working for one hour.700 small cakes will require the equivalent of 20 helpers working for one hour. This means if only one hour were available we would need 30 helpers. But since three hours are available we can use 10helpers.3.Correct Answer: EExplanation:Indian stamps are common to both ratios. Multiply both ratios by factors such that the Indian stamps are represented by the same number. US : Indian = 5 : 2,n n d ian :British =5: 1.Multiply the first by 5,and the second by 2.Now US : Indian = 25 : 10, and Indian : British = 10 : 2 Hence the two ratios can be combined and US : British = 25 : 24.Correct Answer: CExplanation:Draw the diagram.The diagonal of the rectangle is the diameter of the circle. The diagonal is the hypotenuse of a 3,4,5triangle,and is therefore,5. Circumference = π.diameter = 5π5.Correct Answer: DExplanation:If two sets of four consecutive integers have one integer in common,the total in the combined set is 7., and we can write the sets as n + (n + 1) + (n + 2) + (n + 3 ) and (n + 3) + (n + 4) + (n + 5) + (n + 6) Note that each term in the second set is 3more than the equivalent term in the first set.Since there are four terms the total of the differences will be 4x 3=12SAT 考试数学练习题 4来源 : 91SAT 考试网时间 : 2009 年 08 月 20 日1. If f(x) = (x + 2) / (x-2) for all integers except x=2, which of the following has the greatest value?A. f(-1)B. f(0)C. f(1)D. f(3)E. f(4)2. ABCD is a square of side 3, and E and F are the mid points of sides AB and BC respectively. What is the area of the quadrilateral EBFD ?A. 2.25B. 3C. 4D. 4.5E. 63. If n ≠ 0, which of the following mus t be greater than n?I 2nII n²III 2 - nA. I onlyB. II onlyC. I and II onlyD. II and III onlyE. None4. After being dropped a certain ball always bounces back to 2/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (in inches) will it reach after its fourth bounce?A. 20B. 15C. 8D. 5E. 3.25. n and p are integers greater than 15n is the square of a number75np is the cube of a number.The smallest value for n + p isA. 14B. 18C. 20D. 30E. 50答案：1.Correct Answer: DExplanation:You can solve this by back solving – substitute the answer choices in the expression and see which gives the greatest value.satA (-1 + 2) / (-1-2) = -2 / 2 = -1;B (0 + 2) / (0-2) = 2/ -2 = -1;C (1 + 2) / (1-2) = 3/-1 = -3;D (3 + 2) / (3-2) = 5/1 = 5 ;E (4+ 2) / (4-2) = 6/2 = 3If you had just chosen the largest value for x you would have been wrong. So although itlooks a long method, it is actually quick and accurate since the numbers are really simple and you can do the math in your head.2.Correct Answer: DExplanation:(Total area of square - sum of the areas of triangles ADE and DCF) will give the area of the quadrilateral 9 - (2 x ½ x 3 x 1.5) = 4.53.Correct Answer: EExplanation:Remember that n could be positive negative or a fraction. Try out a few cases: In case I, if n is -1, then 2n is less than n. In case II, if n is a fraction such as ½ then n2 will be less than n. Incase III, if n is 2, then 2-n = 0, which is less than n. Therefore, none of the choices must be greaterthan n4.Correct Answer: CExplanation:If after each bounce it reaches 2/5 of the previous height, then after the second bounce it will reach 2/5 x 125. After the third it will reach 2/5 x 2/5 x 125. After the fourth it will reach 2/5 x 2/5 x 2/5 x 125. This cancels down to 2 x 2 x 2 = 85.Correct Answer: AExplanation:The smallest value for n such that 5n is a square is 5. 75np can now be written as 75 x 5 x p. This gives prime factors.... 3 x 5 x 5 x 5 x p To make the expression a perfect cube, p will have to have factors 3 x 3 , and hence p =9 n + p = 5 + 9 = 14SAT 考试数学练习题 5来源 : 太傻网考试频道整理时间 : 2009 年 08 月 27 日1. If f(x) = x² – 3, where x is an integer, which of the following could be a value of f(x)?I 6II 0III -6A. I onlyB. I and II onlyC. II and III onlyD. I and III onlyE. I, II and IIICorrect Answer: A解析:Choice I is correct because f(x) = 6 when x=3. Choice II is incorrect because to make f(x) =0, x² would have to be 3. But 3 is not the square of an integer. Choice III is incorrect because to make f(x) = 0, x² would have to be –3 but squares cannot be negative. (The minimum value for x2 is zero; hence, the minimum value for f(x) = -3)2. For how many integer values of n will the value of the expression 4n + 7 be an integer greater than 1 and less than 200?A. 48B. 49C. 50D. 51E. 52Correct Answer: C解析:1 < 4n + 7 < 200. n can be 0, or -1. n cannot be -2 or any other negative integer or the expression 4n + 7 will be less than1. The largest value for n will be an integer < (200 - 7) /4. 193/4 = 48.25, hence 48. The number of integers between -1 and 48 inclusive is 503. In the following correctly worked addition sum, A,B,C and D represent different digits, and all the digits in the sum are different. What is the sum of A,B,C and D?A. 23B. 22C. 18D. 16E. 14Correct Answer: B解析:First you must realize that the sum of two 2-digit numbers cannot be more that 198 (99 + 99). Therefore in the given problem D must be 1. Now use trial and error to satisfy the sum 5A +BC = 143. A + C must give 3 in the units place, but neither can be 1 since all the digits have to be different. Therefore A + C = 13. With one to carry over into the tens column, 1 + 5 + B = 14, and B = 8. A + C + B + D = 13 + 8 + 1 = 224. 12 litres of water a poured into an aquarium of dimensions 50cm length , 30cm breadth, and 40 cm height. How high (in cm) will the water rise?(1 litre = 1000cm³)A. 6B. 8C. 10D. 20E. 40Correct Answer: B解析:Total volume of water = 12 liters = 12 x 1000 cm3. The base of the aquarium is 50 x 30 = 1500cm3. Base of tank x height of water = volume of water. 1500 x height = 12000; height = 12000 / 1500 = 85. Six years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ?A. 11/P + 6B. P/11 +6C. 17 - P/6D. 17/PE. 11.5PCorrect Answer: A解析:Let Ben’s age now be B. Anita’s age now is A. (A - 6) = P(B - 6)But A is 17 and therefore 11 = P(B - 6). 11/P = B-6(11/P) + 6 = BSAT 考试数学练习题 6来源 : 太傻网考试频道整理时间 : 2009 年 08 月 27 日1. The distance from town A to town B is five miles. C is six miles from B. Which of the following could be the distance from A to C?I 11II 1III 7A. I onlyB. II onlyC. I and II onlyD. II and III onlyE. I, II, or III. Correct Answer: E 解析:Do not assume that AB and C are on a straight line. Make a diagram with A and B marked 5 miles apart. Draw a circle centered on B, with radius 6. C could be anywhere on this circle. The minimum distance will be 1, and maximum 11, but anywhere in between is possible.2. √5 percent of 5√5 =A. 0.05B. 0.25C. 0.5D. 2.5E. 25Correct Answer: B解析:We can write the statement mathematically, using x to mean ‘of’ and /100 for ‘per cent’ . S o ( √5/100) x 5√5 = 5 x 5 /100 = 0.253. If pqr = 1 , rst = 0 , and spr = 0, which of the following must be zero?A. PB. QC. RD. SE. TCorrect Answer: D解析:If pqr = 1, none of these variable can be zero. Since spr = 0 , and since p and r are not zero, s must be zero. (Note that although rst = 0, and so either s or t must be zero, this is not sufficient to state which must be zero)4.A. 1/5B. 6/5C. 6³D. 64 / 5E. 64Correct Answer: E解析:65 = 64x 6(64 x 6) - 64 = 64(6 - 1) = 64 x 5 Now, dividing by 5 will give us 645. -20 , - 16 , - 12 , -8 ....In the sequence above, each term after the first is 4 greater than the preceding term. Which of the following could not be a term in the sequence?A. 0B. 200C. 440D. 668E. 762Correct Answer: E解析:All terms in the sequence will be multiples of 4. 762 is not a multiple of 4SAT 考试数学练习题 7来源 : 太傻网考试频道整理时间 : 2009 年 08 月 27 日1. If a² = 12, then a4 =A. 144B. 72C. 36D. 24E. 16Correct Answer:A解析:a4 = a2 x a2 = 12 x 12 = 1442. If n is even, which of the following cannot be odd?I n + 3II 3nIII n² - 1A. I onlyB. II onlyC. III onlyD. I and II onlyE. I, II and IIICorrect Answer: B解析:In case I , even plus odd will give odd. In case II, odd times even will give even. In case III even squared is even, and even minus odd is odd. (You can check this by using an easy evennumber like 2 in place of n). Only case II cannot be odd.3. One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle?I 24II 20III 5A. I onlyB. II onlyC. III onlyD. II and III onlyE. I, II and IIICorrect Answer: D解析:The maximum area of the triangle will come when the given sides are placed at right angles. If we take 8 as the base and 5 as the height the area = ½ x 8 x 5 = 20. We can alter the angle between the sides to make it less or more than 90, but this will only reduce the area. (Draw it out for yourself). Hence the area can be anything less than or equal to 20.4. A certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 91 pounds?A. 12B. 11C. 10D. 9E. 8Correct Answer: E解析:Food consumed per day = 39/6. In the remaining days it will consume 91 - 39 pounds = 52 pounds. Now divide the food by the daily consumption to find the number of days. 52 / (39/6) = 52 x (6 / 39) = 85. A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?A. 8pB. pqC. pq + 27D. -pE. (p - q)6Correct Answer: C解析:A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5. Consider the answer choices in turn. 8 is the cube of 2, and p is a cube, and so the product will also be a cube. pq will also be a cube as shown above.pq is a cube and so is 27, but their sum need not be a cube. Consider the case where p =1 and q = 8, the sum of pq and 27 will be 35 which has factors 5 x 7 and is not a cube. -p will be a cube. Since the difference between p and q is raised to the power of 6, this expression will be a cube (with cube root = difference squared).SAT 考试数学练习题 8来源 : 太傻网考试频道整理时间 : 2009 年 08 月 27 日6. What is the length of the line segment in the x-y plane with end points at (-2,-2) and (2,3)?A. 3B. √31C. √41D. 7E. 9Correct Answer: C解析:Sketch a diagram and calculate the distance (hypotenuse of a right triangle) using Pythagoras theorem.Vertical height of triangle = 5 ; horizontal side = 4 ; hypotenuse = √(25 + 16) = √417. n is an integer chosen at random from the set{5, 7, 9, 11 }p is chosen at random from the set{2, 6, 10, 14, 18}What is the probability that n + p = 23 ?A. 0.1B. 0.2C. 0.25D. 0.3E. 0.4Correct Answer:A解析:Each of the integers in the first set could be combined with any from the second set, giving a total of 4 x 5 = 20 possible pairs. Of these the combinations that could give a sum of 23 are (5 + 18), and (9 + 14). This means that the probability of getting a sum of 23 is 2/20 = 1/108. A dress on sale in a shop is marked at $D. During the discount sale its price is reduced by 15%. Staff are allowed a further 10% reduction on the discounted price. If a staff member buys the dress what will she have to pay in terms of D ?A. 0.75DB. 0.76DC. 0.765DD. 0.775DE. 0.805DCorrect Answer: C解析:If the price is reduced by 15 %, then the new price will be 0.85D. If this new price is further reduced by 10%, the discounted price will be 0.9 x 0.85D = 0.765D9. All the dots in the array are 2 units apart vertically and horizontally. What is the length of the longest line segment that can be drawn joining any two points in the array without passing through any other point ?A. 2B. 2√2C. 3D. √ 10E. √20Correct Answer: E解析:The longest line segment that can be drawn without passing through any dots other than those at the beginning and end of the segment, such a line could go from the middle dot in the top row to either the bottom left or right dot. In any case the segment will be the hypotenuse of a right triangle with sides 2 and 4. Using Pythagoras theorem the hypotenuse will be √(2 ² + 4 ² ) = √2010. If the radius of the circle with centre O is 7 and the measure of angle AOB is 100, what is the best approximation to the length of arc AB ?A. 9B. 10C. 11D. 12E. 13Correct Answer: D解析:If the radius is 7, the circumference = 14π . The length of the arc is 100/360 of the circumference. Taking π as 22/7 we get. (100 x 14 x 22) / (360 x 7) which reduces to 440/ 36 = 12.22 (i.e. approx. 12)。
Area of a Triangle
A=½ xbxh
A=½ x
x
A= A=
New Example
(-) √ ² C x ÷ 5 6 7 8 9 + - On
0 0 1 2 3 4 . = Ans
15 cm
Aபைடு நூலகம்ea of Obtuse Triangle
2 cm
A=½ xbxh
A=½ x
x
A= A=
New Example
(-) √ ² C x ÷ 5 6 7 8 9 + - On
Area of A Right-Angled Triangle
Learning Intention
1. Use the area of a rectangle formula to help us to come up with a formula to calculate the area of any right-angled triangle.
height
base
Area of Triangle
Area Δ = ½ x base x height
This formula works for all types of triangles
Area of Triangle
2 cm 5 cm
A=½ xbxh
A=½ x 5 x 2
A=
5x2÷2
A=
5
New Example
(-) √ ² C x ÷ 5 6 7 8 9 + - On
0 0 1 2 3 4 . = Ans
Area of a Triangle
6 cm
8 cm
A=½ xbxh
A=½ x
x
英语数学问题解答40题
英语数学问题解答40题1. Tom has 5 apples. He gives 2 apples to his friend. How many apples does Tom have now?A. 2B. 3C. 5D. 7答案：B。
解析：Tom 原本有5 个苹果，给了朋友2 个，5 - 2 = 3，所以Tom 现在还有3 个苹果。
本题主要考查减法运算，涉及的英语数学词汇有“apple（苹果）”“give（给）”。
2. There are 8 pencils in a box. 3 more pencils are put into the box. How many pencils are there in total?A. 5B. 8C. 11D. 14答案：C。
解析：盒子里原本有8 支铅笔，又放入3 支，8 + 3 = 11，总共就有11 支铅笔。
本题考查加法运算，相关词汇有“pencil （铅笔）”“total（总共）”。
3. Mary bought 6 books. Each book costs $5. How much money did Mary spend?A. $11B. $25C. $30D. $35答案：C。
解析：Mary 买了6 本书，每本5 美元，6 × 5 = 30，所以Mary 花了30 美元。
此题为乘法运算，涉及“book（（书）”“cost （花费）”等词汇。
4. A cake is divided into 10 pieces. John ate 3 pieces. How many pieces are left?A. 3B. 7C. 10D. 13答案：B。
解析：蛋糕被分成10 块，John 吃了3 块，10 - 3 = 7，还剩下7 块。
本题涉及减法和“piece（块）”这个词汇。
5. There are 15 students in a class. 5 of them are girls. How many boys are there?A. 5B. 10C. 15D. 20答案：B。
数学(英)词典
Aabacus 算盤absence 無,不存在absolute maximum 絕對極大值absolute minimum 絕對極小值absolute value 絕對值abstraction 抽象(化)absurdity 謬論acceleration 加速,加速度access 存取accounting 計算,會計學accurate to one hundredth 精確到百分之一 accurate value 精確值actual range 值域acute angle 銳角acute-angled triangle 銳角三角形acute triangle 銳角三角形addition 加法addition formula 加法公式addition method 加法addition of sets 集合的加法addition of vectors 向量的加法addition principle 加法原理addition rule 加法規則addition sign 加號addition system 加法系addition table 加法表addition term by term 逐項相加addition theorem 加法定理additional condition 附加條件additional term 附加項adjacent 相鄰的,鄰接的adjacent angle 鄰角adjacent continuous fraction 相鄰連分數 adjacent edge 鄰邊,鄰棱adjacent face 鄰面adjacent point 鄰點adjacent side 鄰邊adjacent small side (相鄰)直角邊adjacent vertex 鄰頂點adjoining 毗鄰的,鄰接的adjoining angle 鄰角advanced mathematics 高等數學affirmation 論斷,命題affirmative 肯定的affirmative acknowledgment 肯定確認affirmative proposition 肯定命題age distribution 年齡分佈algebra 代數學,代數algebraic addition formula 代數加法公式algebraic addition theorem 代數加法定理algebraic coding theory 代數編碼理論algebraic curve of order one 一次代數曲線algebraic curve of order two 二次代數曲線 algebraic equation 代數方程algebraic equivalence 代數等價algebraic expression 代數式algebraic fraction 代數分式,代數分數algebraic function 代數函數algebraic geometry 代數幾何(學)algebraic method for solution 代數解法algebraic number theory 代數數論algebraic polynomial 代數多項式algebraic property 代數性質algebraic relation 代數關係algebraic representation 代數表示algebraic root 代數根algebraic sum 代數和algebraic symbol 代數符號algebraic system 代數系統algebraic value 代數值algebraically equivalence 代數等價algebraically irreducible 代數不可約的algebraist 代數學家algebraization 代數化algorism 演算法,阿拉伯(十進位)計數法algorithm 演算法algorithm of division 除數演算法,帶餘(數)除法 aliquot 整除的,等分的,除得盡的aliquot divisor 整除除數,整除因數aliquot part 整除部分aliquot ratio 單分數alphabet 字母表alphabetic code 字母碼alphabetic coding 字母表編碼alphabetic order 字母表次序alphabetically 按字母表alternance 交錯alternant 交錯式alternate 交錯的,交替的alternate angles 交錯角alternate exterior angles 外錯角alternate interior angles 內錯角altitude 高線,高altitude line 高線altitude of a cone 圓錐的高(線)altitude of a tetrahedron 四面體的高(線)altitude of a triangle 三角形的高(線)altitude theorem 高線定理ambiguous point 歧點ambiguous point theorem 歧點定理amicable numbers 親和數amount 量,總數,本利和amplification 放大amplitude 輻角,振幅,幅度amplitude of an oscillation 振動的振幅analytic(al) 解析的,分析的analytic geometry 解析幾何(學)analytic theory of numbers 解析數論analytic vector 解析向量analytics 分析學angle 角angle at center 圓心角angle at circumference 圓周角angle between two circles 兩圓之間的夾角angle between two edges 兩邊之間的夾角angle between two planes 兩平面之間的夾角angle between two straight lines 兩直線之間的夾角 angle-bisector 角平分線angle constant 常數角angle degree 角度angle hook 直角尺angle of circumference 圓周角angle of contingence (兩條曲線的)切線角angle of depression 俯角angle of elevation 仰角angle of intersection 交角angle of position 方位角angle of rotation 旋轉角angular 角的,角度的angular bisector 角平分線angular boundary point 角邊界點angular degree 角的度數( 60 進制)angular metric 角度量annular 環形的annular domain 環形區域annular region 環形區域annulus 圓環anticlockwise direction 逆時針方向anticlockwise revolution 逆時針旋轉antithesis 逆命題antithetic(al) 對偶的,互斥的,相反的antithetic events 互斥事件,不相容事件antitone 反序antitrigonometric function 反三角函數apex 頂點apex angle 頂角apical angle 頂角Apollonius theorem 阿波羅尼奧斯定理aporia 悖論apothem 邊心距Appell equation 阿佩爾方程Appell function 阿佩爾函數application program 應用程式application software 應用軟體applied mathematics 應用數學appreciation 評價,估計approach 接近,趨向,途徑,方法approximate近似的approximate calculation 近似計算approximate solution 近似解approximate value 近似值Arabic ciphers 阿拉伯數字Arabic number system 阿拉伯記數制 Arabic numerals 阿拉伯數字Arabic system of notation 阿拉伯記數制 arbitrarily great 任意大的arbitrarily small 任意小的arbitrary 任意的arbitrary constant 任意常數arc 弧(又稱"有向邊")arc degree 弧度arc length 弧長arc length of a curve 曲線的弧長arc of a circle 圓弧arc-sine 反正弦arc-sine law 反正弦律arc tangent 反正切arc triangle 圓弧三角形arc-trigonometric function 反三角函數 arch 拱形Archimedean axiom 阿基米德公理Archimedean body 阿基米德(多面)體 area 面積area formula 面積公式area method 面積法area of a circle 圓的面積area of a planar figure 平面圖形的面積area of a polygon 多邊形的面積area of a rectangle 矩形的面積area of a sphere 球面的面積area of a surface 曲面的面積area of a trapezium 梯形的面積area of a triangle 三角形的面積area of an ellipse 橢圓的面積area theorem 面積定理argument domain 定義域arithmetic 算術arithmetic average 算術平均值arithmetic difference 算術差arithmetic division 算術除法arithmetic expression 算術(表達)式arithmetic formula 算術公式arithmetic fraction 算術分數arithmetic fundamental theorem 算術基本定理arithmetic-geometric mean 算術幾何平均arithmetic-geometric progression 算術幾何數列arithmetic-geometric sequence 算術幾何序列arithmetic-geometric series 算術幾何級數arithmetic mean 算術平均,算術中項,等差中項arithmetic-mean geometric-mean inequality 算術平均幾何平均不等式 arithmetic number 算術數arithmetic operation 算術運算arithmetic operator 算術算符arithmetic problem 算術問題arithmetic progression 等差數列,算術數列arithmetic proportion 算術比例arithmetic quotient 算術商arithmetic root 算術根arithmetic sequence 等差序列,算術序列arithmetic series 等差級數,又稱"算術級數"arithmetic sign 算術符號arithmetic sum 算術和arithmetically equivalent 算術等價的arithmetico-harmonic mean 算術調和平均arm 邊arm of an angle 角的邊arrangement排列,佈置arrangement in ascending powers 升冪排列 arrangement in descending powers 降冪排列 arrangement problem 排列問題,佈置問題arrangement with repetitions 有重 排列 array 陣列,陣列array by squares 正方形陣列arris 棱arrival time 到達時arrow 箭頭,矢,射arrow diagram 箭頭圖arrow-head 箭頭ascendant 升的,上升的,遞增的ascendant continued fraction 遞增連分數 ascending 升的,上升的,遞增的ascending powers 升冪ascending sequence 遞增序列,昇冪序列ascending series 升列,遞增列ascent 上升,上升的ascent direction 上升方向associative law 結合律associative law of addition 加法結合律associative law of multiplication 乘法結合律 assumed 假定的assumption 假定ast(e)roid 星形[狀]的astroid 星形線astronomy 天文學asymmetric(al) 非對稱的asymmetric graph 非對稱圖asymmetric relation 非對稱關係asymmetry 非對稱性asymmetry coefficient 非對稱係數asymptote 漸近線asymptote of a curve 曲線的漸近線asymptote of a quadric curve 二次曲線的漸近線 asymptotic center 漸近中心asymptotic circle 漸近圓asymptotic fraction 漸近分數asymptotic line 漸近線asymptotic plane 漸近平面asymptotic point 漸近點asymptotic value 漸近值at most 至多augend 被加數auxiliary 輔助的auxiliary calculation 輔助計算auxiliary circle 輔助圓auxiliary construction 輔助作圖auxiliary line 輔助線auxiliary symbol 輔助符號average 平均(值)average absolute value 平均絕對值average error 平均誤差average value 平均值averaging 平均averaging method 平均方法axes (axis的複數) 軸axial coordinates 軸座標(系)axial plane 軸平面axial symmetry 軸對稱axiom 公理(又稱"公設")axiom of inequality 不等量公理axiom of parallels 平行公理axis 軸axis of a conic 二次曲線的軸axis of abscissas 橫(座標)軸axis of applicates 豎(座標)軸axis of coordinates 坐標軸axis of imaginaries 虛軸axis of ordinates 縱(座標)軸axis of reals 實軸axis of revolution 旋轉軸axis of rotation 旋轉軸axis of similarity 相似軸axis of symmetry 對稱軸axisymmetric 軸對稱的axisymmetric problem 軸對稱問題axoid 軸面Bbalance 平衡,均衡balance condition 平衡條件balanced 平衡的,均衡的balanced die 正常骰子ball 球,球體ball-arithmetic 珠算band 帶,帶狀的band chart 帶狀圖band width 帶寬bar 棒,條線bar chart 線條圖bar graph 條線圖barycenter 重心barycentric 重心的barycentric coordinate system 重心坐標系 base angle 底角base circle 基圓,底圓base number 底數base of a cone 錐的底base of a cylinder 柱的底base of a polyhedron 多面體的底base of a power 冪的底base of a prism 棱柱的底base of a pyramid 棱錐的底base of a trapezium 梯形的底base of a triangle 三角形的底base of logarithm 對數的底base of natural logarithms 自然對數的底 base period 基本周期base point 基點base term 底項basic algebraic operation 基本代數運算basic arithmetic operation 基本算術運算 basic construction problem 基本作圖題basic operation 基本運算basic operator 基本運算符basic units 基本單位bell-shaped 形的bell-shaped curve 狀曲線bell-shaped distribution 狀分佈belong to 屬於bend 彎曲bend point 彎曲點bending 彎曲bending line 彎曲線bending point 極值點bending surface 彎曲面best 最佳的best approximate solution 最佳近似解best approximation 最佳逼近best estimate 最佳估計bet 賭注betweenness 中間性bicircle 雙圓bijection 一一映射,雙射bijective 一一映射,雙射bijective function 一一映射函數,雙射函數 bijective mapping 一一映射,雙射binary 二元的,二進位的,雙態的binary addition 二元制加法binary algebraic operation 二元代數運算binary arithmetic 二進位算術binary bit 二進位位元binary code 二進位碼binary logarithm 以2爲底的對數binary notation 二進位記數法binary number 二進位數字binary number system 二進位數字系binary numeration 二進位binary operation 二元運算binary place 二進位數字位binary point 二進位小數點binary quadratic equation 二元二次方程binary representation 二進位表示binary scale 二進位binary system 二進位binary-to-decimal conversion 二進十進位轉換binary unit 二進位單位binomial 二項式,二項的binomial array 賈憲三角形,楊輝三角形,帕斯卡三角形 binomial coefficients 二項式係數binomial curve 二項式曲線binomial distribution 二項分佈binomial equation 二項方程binomial expansion 二項式展開式binomial formula 二項式公式binomial polynomial 二項式binomial series 二項級數binomial theorem 二項式定理bipartition 二分法birth rate 出生率birth statistics 出生統計bisected chords 平分弦bisecting 二等分bisecting line 二等分線bisecting plane 二等分面bisecting point 中點,二等分點bisection 對分,二等分bisector 平分線,平分面bisector of a dihedral angle 二面角的平分面 bisector of a triangle 三角形的角平分線bisector of an angle 角平分線bisector plane 平分面bisectrix 平分線bisectrix of a triangle 三角形的角平分線 bisectrix of an angle 角的平分線bit 位,比特bitangent 雙切線,公切線bitangent curve 雙切曲線bitangent plane 雙切平面,公切平面bitangential curve 雙切曲線body of revolution 旋轉體body of rotation 旋轉體border 邊界,邊緣bottle 瓶bottom 底,底面bound 界,邊界,約束boundary condition 邊界條件boundary line 邊界線boundary point 邊界點boundary value 邊(界)值boundary value condition 邊值條件bounded 有界的bounded above 上有界的bounded below 下有界的boundless 無界的box 箱,框,抽屜box argument 抽屜原理box principle 抽屜原理brace 花括弧,大括弧bracket 方括號,中括弧bracket operation 方括號運算bracket polynomial 方括號多項式branch 分支branch of a tree 樹的分支branch point 分支點branch point of a curve 曲線的分支點bridge 橋brightness 亮度broken bracket 折括弧broken curve 折曲線broken diagonal 折對角線broken line 折線broken number 分數built-up 拼成butterfly theorem 蝴蝶定理Ccalculating rule 計算法則calculation 計算,演算calculator 計算器calendar 曆法cancellation 消去,相約cancellation law 消去律cancellation law for addition 加法消去律cancellation law for multiplication 乘法消去律 cancelling 消去,抵消cancelling denominator 去分母cant 斜面,傾斜capacity 容量cardinal number 基數cardinality of a set 集合的基數cardioid 心臟線cardioid curve 心臟線carry 進位carry digit 進位數carry-over 進位,借位carrying 進位carrying over 進位Cartesian coordinate system 笛卡兒坐標系 Cartesian geometry 笛卡兒幾何(學)cartogram 統計圖cartography 地圖學casting 投擲casting out 棄去, 去cathetus 直角邊Cauchy inequality 柯西不等式center 中心center circle 中心圓center of a ball 球心center of a circle 圓心center of a conic 二次曲線的中心center of a disk 圓盤的中心center of a graph 圖的中心center of circumcircle 外心center of gravity 重心center of inversion 反演中心center of mass 質心center of reflection 反射中心center of rotation 旋轉中心center of symmetry 對稱中心center of volume 體積中心center point 中心點centesimal 百分制的centesimal fraction 百分數centesimal system 百分制centile 百分位數central angle 圓心角central point 中心點central symmetry 中心對稱centre (=center) 中心centrum 心,中心,中核century 百年,一世紀certain event 必然事件chain 鏈chance event 隨機事件change in sign 符號改變,變號change of base of logarithms 對數底變換change of coordinates 座標變換change of metrics 度量變換change of sign 符號改變,變號change of variables 變數變換check 校驗,驗算check digit 校驗位check formula 校驗公式checking 檢驗checking computations 驗算chess board problem 棋盤問題chest of drawers argument 抽屜原理Chinese Fifteen puzzle 中國十五難題Chinese postman problem 中國郵路問題,中國郵遞員問題Chinese remainder theorem 孫子剩餘定理(又稱"中國剩餘定理") choice axiom 選擇公理choice problem 選擇問題chord 弦chord length 弦長chord of a circle 圓的弦chord of a conic 二次曲線的弦chord of contact 切弦chord theorem 弦定理chordal distance 弦距離cipher 密碼,數位ciphertext 密文ciphertext attack 密文攻擊ciphertext message space 密文消息空間circle 圓,圓周circle geometry 圓幾何(學)circle of contact 切(觸)圓circle of similarity 相似圓circle property 圓性質circle quadrangle 圓四角形circle segment triangle 圓弧段三角形circle transformation 圓變換circuit 回路,線路,環道circuit edge 回路邊circuit graph 回路圖circulant flow 迴圈流circulant graph 迴圈圖circular arc 圓弧circular body 圓形體circular cone 圓錐(體)circular conical surface 圓錐面circular conoid 圓劈錐面circular coordinates 圓座標(系)circular curve 圓形曲線,圓點曲線circular cylinder 圓柱(體)circular cylindrical coordinates 圓柱座標(系) circular cylindrical surface 圓柱面circular disc 圓盤circular geometry 圓幾何(學)circular measure 弧度制circular parts 圓部分circular point 圓點,臍點,虛圓點circular ring 圓環circular sector 圓扇形circular segment 圓弓形,圓弧,圓截口circular truncated cone 圓臺circulating decimal 循環小數circum-angle 周角circumcenter 外心circumcircle 外接圓circumcircle of a triangle 三角形的外接圓circumference 周長,圓周circumradius 外接圓半徑circumscribed circle 外接圓circumscribed cone 外切圓錐circumscribed polygon 外切多邊形circumscribed polyhedron 外切多面體circumscribed prism 外切棱柱circumscribed pyramid 外切棱錐circumscribed quadrilateral 外切四邊形circumscribed sphere 外接球面circumscribed triangle 外切三角形circumsphere 外接球面cissoid 蔓葉線claim 斷言class 類,班,組class width 組距classical algebra 經典代數(學)classical algebraic geometry 經典代數幾何(學) classical concept of probability 古典概率概念classification 分類clearing of fractions 去分母cleartext 明文clopen set 閉開集close 近的,鄰近的closed 閉的closed-and-open set 閉開集closed-and-open subset 閉開子集closed arc 閉弧closed boundary 閉邊界closed curve 閉曲線closed-open set 閉開集closed path 閉路closed plane 閉平面closed point 閉點closed ray 閉射線closed region 閉區域closed set 閉集closed subset 閉子集coangle 餘角coarse estimate 粗估計coaxial 共軸的coaxial circles 共軸圓coaxial planes 共軸平面coaxial system of circles 共軸圓系coded system 編碼系統coding 編碼codirection 同向codirectional vectors 同向向量coefficient 係數coefficient of proportionality 比例係數 cofunction 餘函數collection of like terms 合併同類項collection of numbers 數集collection of points 點集collection of sets 集合的集合,集族collinear 共線的collinear lines 共線直線collinear points 共線點collinear vectors 共線向量collision frequency 碰撞頻率collocation 配置,配位元,排列colon 冒號,兩點空間coloring 著色coloring of a graph 圖的著色coloring of a node 結點的著色coloring of a vertex 頂點的著色coloring of an edge 邊的著色coloring problem 著色問題column 列column average 列平均column chart 條形圖column graph 列圖column of a matrix 矩陣的列column sum 列和column total 列總和combination組合combination analysis 組合分析combination law 組合律combination method 組合方法combination number 組合數combinatorics 組合學comma 逗號,逗點command 指令,命令commensurable 可公度的,可通約的commensurable fraction 有理分數commensurable line segments 可公度線段commensurable magnitudes 可公度量commensurable quantities 可公度量commensurable segments 可公度線段common 普通的,公共的common chord 公共弦common denominator 公分母common difference 公差common divisor 公因數,公約數,公因數,公因式 common factor 公因數,公因式common fraction 普通分數,簡分數common logarithm 常用對數common measure 公度,公測度,公因數common multiple 公倍數,公倍式common part 公共部分,交common perpendicular 公垂線common plane 公共面common point 公共點common ratio 公比common root 公根common tangent 公切線common term 通項commutative law 交換律commutative law of addition 加法交換律commutative law of multiplication 乘法交換律 comparability 可比較性comparability graph 可比較圖comparability relation 可比較關係comparability theorem 可比較定理comparison 比較,對比compass construction 圓規作圖法complanar 共面的complanar line 共面直線complement 補,餘,補集,補元complement angle 餘角complement of a number 餘數complement of a set 補集complement of a subset 補子集complementary angle 餘角complementary factor 餘因數complementary number 餘數,補數complementary set 補集,餘集complete square 完全平方completeness 完全性(曾用名"完備性") completeness axiom 完全性公理completeness theorem 完全性定理complex coefficient 複係數complex conjugate root 複共軛根complex coordinates 複座標(系)complex fraction 繁分數complex fractional expression 繁分式complex number 複數complex number plane 複數平面complex plane 複平面componendo 合比定理componendo and dividend 合分比定理 component 分支,分量,連通分支component of a vector 向量的分量component vector 分向量composite number 合數composition and division 合分比公式composition factor 合成因數computation計算computation error 計算誤差computing rule 計算規則,計算尺computing table 計算表concave angle 凹角,優角concave polyhedron 凹多面體conchoid 蚌線concircular 共圓的conclusion 結論concrete number 名數concurrent 共點的,並行的concurrent circles 共點圓concurrent lines 共點線concurrent planes 共點面concyclic 共圓的concyclic points 共圓點condition 條件condition equation 條件方程condition inequality 條件不等式conditional probability 條件概率cone 錐(面),錐(體)cone of the second order 二次錐面cone-point 錐的頂點confocal 共焦的confocal central conics 共焦中心二次曲線confocal central surfaces 共焦中心曲面confocal cones 共焦錐面confocal conic sections 共焦圓錐曲線confocal conics 共焦二次曲線confocal curves 共焦曲線confocal ellipses 共焦橢圓confocal ellipsoids 共焦橢球面confocal parabolas 共焦抛物線confocal paraboloids 共焦抛物面confocal quadrics 共焦二次曲面confocal surfaces 共焦曲面congruence 同 式,同 方程,同 ,合同,全等,疊合,相合,線彙 congruence (in algebra) (代數學中的)合同congruence (in geometry) (幾何學中的)全等congruence axiom 合同公理,全等公理congruence of angles 角的全等congruence of figures 圖形的全等congruence of segments 線段的合同congruence of tangents 切線的合同congruence of triangles 三角形的全等congruence problem 同餘問題congruence theorem 全等定理congruence theory 同餘理論,全等理論congruency 同 ,合同,全等congruent angles 全等角congruent expression 同餘式congruent figures 全等圖形congruent modulo m 模 m 同餘的congruent numbers 同餘數congruent points 相合點,疊合點congruent polyhedral angles 全等多面角congruent polynomial 同餘多項式congruent segments 全等線段,相合線段congruent triangles 全等三角形congruous number 同餘數conic 二次曲線,圓錐曲線,(=conical) 圓錐的 conic node 圓錐頂點conic point 圓錐頂點conic section 二次曲線,圓錐曲線conical curve 圓錐曲線conicoid 二次曲面conjecture 猜想conjugate roots 共軛根connected graph 連通圖consecutive 相鄰的consecutive integers 相鄰整數consecutive numbers 相鄰數consecutive points 相鄰點consecutive primes 相鄰 數consecutive roots 相鄰根consequent 後項conservation 守恒consistence 相容consistency 相容性(曾用名"協調性","和諧性")consistency condition 相容性條件constant 常數,常量,常項,常元constant factor 常數因數constant number 常數constant of proportionality 比例常數constant quantity 常量constant term 常數項,常項constructibility 可構造性,可構成性constructibility axiom 可構造性公理construction 作圖,構造construction problem 作圖題construction with ruler and compasses 尺規作圖法 consumer price index 消費物價指數contact 切觸,接觸,觸點contact circle 切(觸)圓containing 包含的content 庫容,容度,容量contingency 相切,擬切,列聯continuation 延拓,連續continued fraction 連分數,連分式continued fraction approximation 連分式逼近continued fraction expansion 連分數展開continued fraction representation 連分數表示continued multiplication 連乘continued product 連乘積continued proportion 連比continued ratio 連比continuous sum 連續和contour line 等高線,等值線contour map 等高線圖,等值線圖contradiction 矛盾,矛盾式contradictory proposition 矛盾命題contrapositive sentence 逆否命題contrapositive theorem 逆否定理convergent 收斂的convergent almost everywhere 幾乎處處收斂的convergents 漸近分數convergents of continued fraction 連分數的漸近分數 converse domain 值域,後域converse-negative proposition 逆否命題converse of a contrapositive 逆否命題的逆,否命題 converse of an implication 逆蘊涵,逆命題converse proposition 逆命題converse relation 逆關係converse sentence 逆命題converse statement 逆命題converse theorem 逆定理convex 凸,凸的convex angle 凸角,劣角convex figure 凸(圖)形convex polygon 凸多邊形convex polyhedron 凸多面體coordinate 座標coordinate axis 坐標軸coordinate geometry 座標幾何(學)coordinate line 座標線coordinate of a point 點的座標coordinate origin 座標原點coordinate plane 座標平面coordinate subspace 座標子空間coordinate system 坐標系coordinate transformation 座標變換coordinate vector 座標向量coplanar 共面的coplanar lines 共面線coplanar vectors 共面向量corner 隅角,頂點,角點corner point 角點corner polyhedron 頂點多面體corollary 系,推論correct 適定的,適當的,正確的correct to 精確到correcting code 校正碼,改錯碼corresponding angles 同位角corresponding points 對應點corresponding segments 對應線段corresponding sides 對應邊corresponding term 對應項corresponding vertices 對應頂點cosecant 割cosine 弦cosine formula 弦公式cosine function 弦函數cosine theorem 弦定理cotangent 餘切cotangent function 餘切函數countable 可數的counterclockwise 反時針counterclockwise direction 反時針方向 counter-example 反例counting 計數counting constants 計數常數countless 不可數的covering 覆蓋,覆疊critical behavior 臨界狀態critical path 關鍵路線critical phenomenon 臨界現象critical point 臨界點critical value 臨界值critical vertex 臨界(頂)點cross 交叉,互cross cut 截線,交cross figure 截面圖cross multiplication 叉乘(法)cross point 交叉點cross ratio 交比cross section 截面,截痕crossed multiplication 交叉乘法crossed product 叉積,向量積crossfooting 交叉驗算crossing point 交點crosswise 交叉的,十字狀的cruciform 十字形,十字架形cryptanalysis 密碼分析crypto-protocol 密碼協定cryptogram 密報,密碼cryptography 密碼學crystal 晶體cubature formula 求體積公式cube 立方,立方體cube of a number 數的立方cube root 立方根cube root test 立方根檢驗cubic(al) 三次的,立方的cubic binary quantic 二元三次形式,二元三次型 cubic equation 三次方程cubic equation with one unknown 一元三次方程 cubic function 三次函數cubic lattice 立方格cubic number 立方數cubic parabola 三次抛物線cubic polynomial 三次多項式cubic remainder 開立方餘數cubic root 立方根cubic sum 立方和cuboid 長方體curvature 曲率curvature of a curve 曲線的曲率curve 曲線curve arc 曲線弧curve of normal distribution 正態分佈曲線curve of second order 二次曲線(又稱"圓錐曲線") curve of steepest descent 最速降線curve of the tangent 正切曲線curved 曲線的,彎曲的cut line 割線cycloid 擺線,旋輪線cylinder (圓)柱,(圓)柱體,柱面cylindrical surface 柱面cylindroid 柱形面cypher 數位,密碼,零Ddata 資料data analysis 資料分析data base 資料庫data compression 資料壓縮data encryption 資料加密data model 資料模型data processing 資料處理decade 十進位數字,十進位的decadic number 十進位數字,小數decagon 十邊形decagonal number 十邊形數decagram 十角星decahedron 十面體decile 十分位數decimal (十進位)小數,十進位的decimal approximation of a real number 實數的小數近似 decimal arithmetic 小數算術decimal-binary 十進-二進位的decimal-binary conversion 十進-二進位轉換decimal carry 十進位進位decimal computation system 十進位計算體系decimal digit 十進位數字字decimal expansion 十進位展開decimal fraction 十進位小數decimal logarithm 十進對數,常用對數decimal notation 十進位記數法decimal number 十進位數字,小數decimal number system 十進位數字制decimal numeration 十進位記數法decimal part 小數部分decimal place 小數位decimal point 小數點decimal scale 十進位decimal system 十進位deck (卡片)組,紙牌decomposable into factors 可分解成因數的decomposed prime number 分解 數decomposition into factors 因數分解decomposition of a fraction 分數的分解decomposition problem of integers 整數分拆問題decreasing (遞)減的decreasing progression 遞減數列,遞減級數decreasing rate (遞)減率decreasing sequence 遞減序列decreasing series 遞減級數deduction 演繹,推理deduction reasoning 演繹推理deduction rule 演繹法則deduction theorem 演繹定理deductive method 演繹法deductive proof 演繹證明deductive reasoning 演繹推理defect 虧數,虧量,虧值defect number 虧數defective 虧損defective number 虧數defective unit 缺項deferent 圓心軌迹deficient number 虧數,不足數deficit 不足,赤字defined value 定義值definite integral 定積分definition 定義definitional domain 定義域degenerate case 退化情況degenerate conic 退化二次曲線degenerate conic sections 退化圓錐曲線degenerate surface of the second order 退化二次曲面 degree 度,次,次數,階,級degree (measure of an angle) 度(角的度量) degree measure 角度(法)degree of a polynomial 多項式的次數degree of an angle 角的度數degree of an equation 方程的次數degree of freedom 自由度degree of precision 精確度degree of stability 穩定度degree relation 次數關係degression 累減,遞減delay 延遲deletion 捨棄(運算),刪除delicate 銳角的,尖銳的deltoid 箏形,矛尖形demand 需求demand curve 需求曲線demiarc 半弧demonstration 證明denary 十的,十進位的denary logarithm 以 10 爲底的對數,常用對數 denary scale 十進位計數法denial 否定denominate number 名數denomination 命名,名稱denominator 分母denominator divisor 分母因數density 密度,稠密性density estimator 密度估計量density function 密度函數density of a distribution 分佈密度density of a set 集合的密度denumerability 可數性denumerable 可數的denumerable number 可數數denumerant 整數分拆數denumeration 枚舉,計數departure 偏差,偏離depreciation 減少,降低,貶值depression 降階,降次depth 深度derangement 更列,重排derivation 求導,導子,推導derivative 導數,微商derivative formula 導數公式derivative of order n n階導數descendant 後繼,後項descending 降的,下降的,遞減的descending continued fraction 遞減連分數 descending induction 遞減歸納法descending order of power 降冪次序descending power series 降冪級數descending powers 降冪descending sequence 遞減序列descending series 降列,遞減列descent 下降descent method 下降法descriptive 描述的,描述性的descriptive geometry 畫法幾何(學)design 設計desired value 期望值determinacy 確定性determinacy axiom 確定性公理determinant 行列式determinant of the coefficients 係數行列式 determinant rank 行列式秩determinantal divisor 行列式因數determinantal equation 行列式方程determinantal expansion 行列式展開式determinantal polynomial 行列式多項式 determinantal representation 行列式表示 determinating condition 決定條件determination 決定,確定development 展開,發展deviation 偏差,離差,變差deviation from an estimate 估計偏差deviation from the mean 平均偏差dextral set 右手系diagonal 對角線,對角的,對角線的diagonal line 對角線diagonal of a polygon 多邊形的對角線diagonal of a polyhedron 多面體的對角線 diagonal plane 對角面diagonal point 對角點diagonal principle 對角線原理diagonal sum 對角和diagonal term 對角項diagonal triangle 對角三角形diagram 圖,圖形,圖式,圖表diagram method 圖解法diameter 直徑diameter of a circle 圓的直徑diameter of a conic 二次曲線的直徑diameter of a sphere 球面的直徑diametric line 直徑diametric plane 直徑面diametric surface 直徑曲面diamond 菱形,鑽石dib 籌碼dictionary order 字典序didactics of mathematics 數學教學法die 骰子difference of partial sums 部分和的差difference of sets 差集difference proportion 差比,算術比different from zero 異於零differential 微分differential equation 微分方程differential quotient 導數,微商digit 數位,數位,位元(數)digit positional 數位digital 數位的。
高中必修数学知识点总结及公式大全
高中必修数学知识点总结及公式大全1.二次函数的标准形式为y=ax^2+bx+c。
The standard form of a quadratic function is y=ax^2+bx+c.2.一次函数的标准形式为y=kx+b。
The standard form of a linear function is y=kx+b.3.三角函数sin、cos、tan分别表示正弦、余弦、正切。
The trigonometric functions sin, cos, tan represent sine, cosine, tangent respectively.4.三角函数的周期性是它们的重要特征之一。
The periodicity of trigonometric functions is one oftheir important characteristics.5.平行四边形的面积公式为S=底×高。
The formula for the area of a parallelogram isS=base×height.6.直角三角形的勾股定理为a^2 + b^2 = c^2。
The Pythagorean theorem for a right-angled triangle isa^2 + b^2 = c^2.7.两点间距离公式为d=sqrt[(x2-x1)^2 + (y2-y1)^2]。
The distance formula between two points is d=sqrt[(x2-x1)^2 + (y2-y1)^2].8.二次方程的解法包括用公式法和配方法。
The methods for solving quadratic equations include using the formula and completing the square.9.函数奇偶性的判定方法是f(-x) = f(x)或f(-x) = -f(x)。
英语数学试题及答案
英语数学试题及答案一、选择题（每题2分，共20分）1. What is the value of \( x \) in the equation \( 2x + 5 =17 \)?A. 3B. 6C. 8D. 102. If the perimeter of a rectangle is 24 cm and its length is8 cm, what is the width?A. 4 cmB. 6 cmC. 8 cmD. 10 cm3. The sum of two numbers is 15 and their difference is 5. What is the larger number?A. 5B. 7C. 10D. 154. What is the area of a triangle with a base of 10 cm and a height of 5 cm?A. 25 cm²B. 30 cm²C. 50 cm²D. 100 cm²5. If a car travels at a speed of 60 km/h for 2 hours, howfar does it travel?A. 60 kmB. 90 kmC. 120 kmD. 150 km6. The product of two numbers is 36. If one number is 4, what is the other number?A. 6B. 8C. 9D. 127. What is the value of \( y \) in the equation \( y - 3 = 12 \)?A. 9B. 10C. 15D. 188. The average of three numbers is 10. If two of the numbers are 8 and 12, what is the third number?A. 10B. 12C. 14D. 169. If 20% of a number is 100, what is 60% of the same number?A. 300B. 400C. 500D. 60010. The ratio of boys to girls in a class is 3:2. If thereare 24 girls, how many boys are there?A. 18B. 24C. 36D. 48二、填空题（每题3分，共15分）11. Solve the equation \( 3x - 7 = 14 \). The value of \( x \) is _______.12. The circumference of a circle is \( 2\pi r \), where \( r \) is the radius. If the circumference is 44 cm, the radiusis _______.13. If the area of a square is 64 cm², the length of eachside is _______.14. The volume of a cube is given by \( V = s^3 \), where\( s \) is the side length. If the volume is 125 cm³, theside length is _______.15. The sum of three consecutive even numbers is 48. The middle number is _______.三、简答题（每题5分，共30分）16. A rectangular garden is 50 meters long and 30 meters wide. Calculate the area of the garden.17. A company produces 300 units of a product every day. If each unit sells for $10, calculate the daily revenue.18. If a car decreases its speed by 20 km/h, its travel time for a certain distance increases by 10 minutes. If theoriginal speed was 60 km/h, what is the distance?19. A school has 480 students. If 40% of the students are in the science club, how many students are in the science club?20. A triangle has sides of lengths 5 cm, 12 cm, and 13 cm.Is it a right triangle? Explain why or why not.四、解答题（每题10分，共35分）21. A ladder placed against a wall makes a 75° angle withthe ground. If the foot of the ladder is 4 meters away fromthe wall, how long is the ladder?22. A bank account earns simple interest. If $2000 is deposited and the interest rate is 5% per annum, calculatethe total amount after 3 years.23. A manufacturer produces two types of products: Type A and Type B. The profit from Type A is $5 per unit, and from TypeB is $10 per unit. If the manufacturer produces 100 units of Type A and 200 units of Type。
第十一章三角形知识点归纳
第十一章三角形知识点归纳Chapter 11: Summary of Triangle KnowledgeTopic 1: XXX1.The sum of two sides of a triangle is greater than the third side.2.The difference een two sides of a triangle is less than the third side.3.The minimum sum of two sides is greater than the third side to form a triangle.4.Given the lengths of two sides of a triangle (a and b)。
the range of the third side is: the difference een the two sides is less than the third side。
which is less than the sum of the two sides。
XXX: Which of the following sets of line segments can form a triangle。
A。
5.6.10 B。
5.6.11 C。
3.4.8 D。
4.4.8Example: Given two sides of a triangle as 7 and 12.the range of the third side is:Topic 2: XXXXXX: Which of the following shapes are stable。
A。
XXX-angled triangleTopic 3: Heights of TrianglesA perpendicular line XXX side BC。
and the n point is D。
北师大版五年级上册数学知识点汇总
北师大版五年级上册数学知识点汇总北师大版五年级上册数学知识点汇总第一单元小数除法1、除数是整数的小数除法计算法则：按照整数除法的法则去除，商的小数点要和被除数的小数点对齐。
如果除到被除数的末尾仍有余数，就在余数后面添再继续除。
2、除数是小数的小数除法计算法则：先移动除数的小数点，使它变成整数；除数的小数点向右移动几位，被除数的小数点也向右移动几位（位数不够的，在被除数末尾用补足），然后按照除数是整数的小数除法进行计算。
3、在小数除法中的发现：①当除数大于1时，商小于被除数。
例如：3.5÷5=0.7②当除数小于1时，商大于被除数。
例如：3.5÷0.5=74、小数除法的验算方法：①商×除数=被除数（通用）②被除数÷商=除数5、商的近似数：根据要求要保留的小数位数，决定商要除出几位小数，再根据“四舍五入”法保留一定的小数位数，求出商的近似数。
例如：要求保留一位小数的，商除到第二位小数可停下来；要求保留两位小数的，商除到第三位小数停下来……如此类推。
6、循环小数问题：A、小数部分的位数是有限的小数，叫做有限小数。
例如，0.37、1.4135等。
B、小数部分的位数是无限的小数，叫做无限小数。
例如5.3… 7.…等。
C、一个数的小数部分，从某位起，一个数字或者几个数字依次不断重复出现，这样的小数叫做循环小数。
例如5.3… 3.… 5.7171…D、一个循环小数的小数部分，依次不断重复的数字，叫做小数的循环节。
例如5.333…的循环节是3，4.6767…的循环节是67，6.xxxxxxx…的循环节是258.7、用简便方法写循环小数的方法：只写一个循环节，并在这个循环节的首位和末位上面记一个小圆点。
只有一个数字循环节的，就在这个数字上面记一个小圆点。
有两位小数循环的，就在这两位数字上面，记上小圆点。
有三位或以上小数循环的，在首位和末位记上小圆点。
8、除法中的变化规律：①商不变性质：被除数和除数同时扩大或缩小相同的倍数（0除外），商不变。
sat数学试题
1、 Cone A has volume 24. When its radius and height are multiplied by the same factor， the cone’s surface area doubles. What is Cone A's new volume?
（A)
5。 If the radius of the circle with centre O is 7 and the measure of angle AOB is 100, what is the best approximation to the length of arc AB ?
A. 9
B。 10 C. 11 D. 12 E。 13 Correct Answer: D 解析: If the radius is 7, the circumference = 14π. The length of the arc is 100/360 of the circumference。 Taking π as 22/7 we get。（100 x 14 x 22） / （360 x 7） which reduces to 440/ 36 = 12。22 （i。e。 approx。 12)
曼昆经济学原理英文版文案加习题答案8章
144WHAT’S NEW IN THE S EVENTH EDITION: A new In the News box on ―The Tax Debate ‖ has been added.LEARNING OBJECTIVES: By the end of this chapter, students should understand:how taxes reduce consumer and producer surplus.the meaning and causes of the deadweight loss from a tax.why some taxes have larger deadweight losses than others.how tax revenue and deadweight loss vary with the size of a tax.CONTEXT AND PURPOSE:Chapter 8 is the second chapter in a three-chapter sequence dealing with welfare economics. In theprevious section on supply and demand, Chapter 6 introduced taxes and demonstrated how a tax affects the price and quantity sold in a market. Chapter 6 also described the factors that determine how the burden of the tax is divided between the buyers and sellers in a market. Chapter 7 developed welfare economics —the study of how the allocation of resources affects economic well-being. Chapter 8 combines the lessons learned in Chapters 6 and 7 and addresses the effects of taxation on welfare. Chapter 9 will address the effects of trade restrictions on welfare. The purpose of Chapter 8 is to apply the lessons learned about welfare economics in Chapter 7 to the issue of taxation that was addressed in Chapter 6. Students will learn that the cost of a tax to buyers and sellers in a market exceeds the revenue collected by the government. Students will also learn about the factors that determine the degree by which the cost of a tax exceeds the revenue collected by the government.8 APPLICATION: THE COSTS OF TAXATIONChapter 8 /Application: The Costs of Taxation ❖145KEY POINTS:∙ A tax on a good reduces the welfare of buyers and sellers of the good, and the reduction in consumer and producer surplus usually exceeds the revenue raised by the government. The fall in total surplus—the sum of consumer surplus, producer surplus, and tax revenue—is called thedeadweight loss of the tax.∙Taxes have deadweight losses because they cause buyers to consume less and sellers to produce less, and these changes in behavior shrink the size of the market below the level that maximizes total surplus. Because the elasticities of supply and demand measure how much market participantsrespond to market conditions, larger elasticities imply larger deadweight losses.∙As a tax grows larger, it distorts incentives more, and its deadweight loss grows larger. Because a tax reduces the size of a market, however, tax revenue does not continually increase. It first rises with the size of a tax, but if the tax gets large enough, tax revenue starts to fall.CHAPTER OUTLINE:I. The Deadweight Loss of TaxationA. Remember that it does not matter who a tax is levied on; buyers and sellers will likely share inthe burden of the tax.B. If there is a tax on a product, the price that a buyer pays will be greater than the price the sellerreceives. Thus, there is a tax wedge between the two prices and the quantity sold will be smaller if there was no tax.146 ❖ Chapter 8 /Application: The Costs of TaxationC. How a Tax Affects Market Participants1. We can measure the effects of a tax on consumers by examining the change in consumersurplus. Similarly, we can measure the effects of the tax on producers by looking at the change in producer surplus.2. However, there is a third party that is affected by the tax —the government, which gets total tax revenue of T × Q. If the tax revenue is used to provide goods and services to the public,then the benefit from the tax revenue must not be ignored.3. Welfare without a Taxa. Consumer surplus is equal to: A + B + C.b. Producer surplus is equal to: D + E + F.c. Total surplus is equal to: A + B + C + D + E + F.4. Welfare with a Taxa. Consumer surplus is equal to: A.Chapter 8 /Application: The Costs of Taxation ❖147b. Producer surplus is equal to: F.c. Tax revenue is equal to: B + D.d. Total surplus is equal to: A + B + D + F.5. Changes in Welfarea. Consumer surplus changes by: –(B + C).b. Producer surplus changes by: –(D + E).c. Tax revenue changes by: +(B + D).d. Total surplus changes by: –(C + E).6. Definition of deadweight loss: the fall in total surplus that results from a marketdistortion, such as a tax.D. Deadweight Losses and the Gains from Trade1. Taxes cause deadweight losses because they prevent buyers and sellers from benefiting fromtrade.2. This occurs because the quantity of output declines; trades that would be beneficial to boththe buyer and seller will not take place because of the tax.3. The deadweight loss is equal to areas C and E (the drop in total surplus).4. Note that output levels between the equilibrium quantity without the tax and the quantitywith the tax will not be produced, yet the value of these units to consumers (represented by the demand curve) is larger than the cost of these units to producers (represented by thesupply curve).148 ❖ Chapter 8 /Application: The Costs of TaxationII. The Determinants of the Deadweight LossA. The price elasticities of supply and demand will determine the size of the deadweight loss that occurs from a tax.1. Given a stable demand curve, the deadweight loss is larger when supply is relatively elastic.2. Given a stable supply curve, the deadweight loss is larger when demand is relatively elastic. B. Case Study: The Deadweight Loss Debate1. Social Security tax and federal income tax are taxes on labor earnings. A labor tax places a tax wedge between the wage the firm pays and the wage that workers receive.2. There is considerable debate among economists concerning the size of the deadweight lossfrom this wage tax.3. The size of the deadweight loss depends on the elasticity of labor supply and demand, andthere is disagreement about the magnitude of the elasticity of supply.Chapter 8 /Application: The Costs of Taxation ❖149a. Economists who argue that labor taxes do not greatly distort market outcomes believethat labor supply is fairly inelastic.b. Economists who argue that labor taxes lead to large deadweight losses believe that laborsupply is more elastic.III. Deadweight Loss and Tax Revenue as Taxes VaryA. As taxes increase, the deadweight loss from the tax increases.B. In fact, as taxes increase, the deadweight loss rises more quickly than the size of the tax.1. The deadweight loss is the area of a triangle and the area of a triangle depends on thesquare of its size.150 ❖ Chapter 8 /Application: The Costs of Taxation2. If we double the size of a tax, the base and height of the triangle both double so the area ofthe triangle (the deadweight loss) rises by a factor of four.C. As the tax increases, the level of tax revenue will eventually fall.D. Case Study: The Laffer Curve and Supply-Side Economics1. The relationship between the size of a tax and the level of tax revenues is called a Laffercurve.2. Supply-side economists in the 1980s used the Laffer curve to support their belief that a drop in tax rates could lead to an increase in tax revenue for the government.3. Economists continue to debate Laffer’s argument.a. Many believe that the 1980s refuted Laffer’s theory.b. Others believe that the events of the 1980s tell a more favorable supply-side story.c. Some economists believe that, while an overall cut in taxes normally decreases revenue,some taxpayers may find themselves on the wrong side of the Laffer curve. E. In the News: The Tax Debate1. Recently, policymakers have debated the effects of increasing the tax rate, particularly onhigher-income taxpayers.2. These two opinion pieces from The Wall Street Journal present both sides of the issue. ALTERNATIVE CLASSROOM EXAMPLE: Draw a graph showing the demand and supply of paper clips. (Draw each curve as a 45-degree line so that buyers and sellers will share any tax equally.) Mark the equilibrium price as $0.50 (per box) and the equilibrium quantity as 1,000 boxes. Show students the areas of producer and consumer surplus.Impose a $0.20 tax on each box. Assume that sellers are required to ―pay‖ the tax to the government. Show students that:▪ the price buyers pay will rise to $0.60.▪ the price sellers receive will fall to $0.40.▪ the quantity of paper clips purchased will fall (assume to 800 units).▪ tax revenue would be equal to $160 ($0.20 800).Have students calculate the area of deadweight loss. (You may have to remind students how to calculate the area of a triangle.)Show students that as the tax increases (to $0.40, $0.60, and $0.80), tax revenue rises and then falls, and the deadweight loss increases.Chapter 8 /Application: The Costs of Taxation ❖151152 ❖ Chapter 8 /Application: The Costs of TaxationSOLUTIONS TO TEXT PROBLEMS:Quick Quizzes1. Figure 1 shows the supply and demand curves for cookies, with equilibrium quantity Q 1 andequilibrium price P 1. When the government imposes a tax on cookies, the price to buyers rises to P B , the price received by sellers declines to P S , and the equilibrium quantity falls to Q 2. The deadweight loss is the triangular area below the demand curve and above the supply curve between quantities Q 1 and Q 2. The deadweight loss shows the fall in total surplus that results from the tax.Figure 12. The deadweight loss of a tax is greater the greater is the elasticity of demand. Therefore, a tax on beer would have a larger deadweight loss than a tax on milk because the demand forbeer is more elastic than the demand for milk.Chapter 8 /Application: The Costs of Taxation ❖1533. If the government doubles the tax on gasoline, the revenue from the gasoline tax could riseor fall depending on whether the size of the tax is on the upward or downward slopingportion of the Laffer curve. However, if the government doubles the tax on gasoline, you canbe sure that the deadweight loss of the tax rises because deadweight loss always rises as thetax rate rises.Questions for Review1. When the sale of a good is taxed, both consumer surplus and producer surplus decline. Thedecline in consumer surplus and producer surplus exceeds the amount of governmentrevenue that is raised, so society's total surplus declines. The tax distorts the incentives ofboth buyers and sellers, so resources are allocated inefficiently.2. Figure 2 illustrates the deadweight loss and tax revenue from a tax on the sale of a good.Without a tax, the equilibrium quantity would be Q1, the equilibrium price would be P1,consumer surplus would be A + B + C, and producer surplus would be D + E + F. Theimposition of a tax places a wedge between the price buyers pay, P B, and the price sellersreceive, P S, where P B = P S + tax. The quantity sold declines to Q2. Now consumer surplus isA, producer surplus is F, and government revenue is B + D. The deadweight loss of the tax isC+E, because that area is lost due to the decline in quantity from Q1 to Q2.Figure 23. The greater the elasticities of demand and supply, the greater the deadweight loss of a tax.Because elasticity measures the responsiveness of buyers and sellers to a change in price,higher elasticity means the tax induces a greater reduction in quantity, and therefore, agreater distortion to the market.4. Experts disagree about whether labor taxes have small or large deadweight losses becausethey have different views about the elasticity of labor supply. Some believe that labor supplyis inelastic, so a tax on labor has a small deadweight loss. But others think that workers canadjust their hours worked in various ways, so labor supply is elastic, and thus a tax on laborhas a large deadweight loss.5. The deadweight loss of a tax rises more than proportionally as the tax rises. Tax revenue,however, may increase initially as a tax rises, but as the tax rises further, revenue eventuallydeclines.Quick Check Multiple Choice1. a2. b3. c4. a5. b6. aProblems and Applications1. a. Figure 3 illustrates the market for pizza. The equilibrium price is P1, the equilibriumquantity is Q1, consumer surplus is area A + B + C, and producer surplus is area D + E +F. There is no deadweight loss, as all the potential gains from trade are realized; totalsurplus is the entire area between the demand and supply curves: A + B + C + D + E +F.Figure 3b. With a $1 tax on each pizza sold, the price paid by buyers, P B, is now higher than theprice received by sellers, P S, where P B = P S + $1. The quantity declines to Q2, consumersurplus is area A, producer surplus is area F, government revenue is area B + D, anddeadweight loss is area C + E. Consumer surplus declines by B + C, producer surplusdeclines by D + E, government revenue increases by B + D, and deadweight lossincreases by C + E.c. If the tax were removed and consumers and producers voluntarily transferred B + D tothe government to make up for the lost tax revenue, then everyone would be better offthan without the tax. The equilibrium quantity would be Q1, as in the case without thetax, and the equilibrium price would be P1. Consumer surplus would be A + C, becauseconsumers get surplus of A + B + C, then voluntarily transfer B to the government.Producer surplus would be E + F, because producers get surplus of D + E + F, thenvoluntarily transfer D to the government. Both consumers and producers are better offthan the case when the tax was imposed. If consumers and producers gave a little bitmore than B + D to the government, then all three parties, including the government,would be better off. This illustrates the inefficiency of taxation.2. a. The statement, "A tax that has no deadweight loss cannot raise any revenue for thegovernment," is incorrect. An example is the case of a tax when either supply or demand is perfectly inelastic. The tax has neither an effect on quantity nor any deadweight loss,but it does raise revenue.b. The statement, "A tax that raises no revenue for the government cannot have anydeadweight loss," is incorrect. An example is the case of a 100% tax imposed on sellers.With a 100% tax on their sales of the good, sellers will not supply any of the good, sothe tax will raise no revenue. Yet the tax has a large deadweight loss, because it reduces the quantity sold to zero.3. a. With very elastic supply and very inelastic demand, the burden of the tax on rubberbands will be borne largely by buyers. As Figure 4 shows, consumer surplus declinesconsiderably, by area A + B, but producer surplus decreases only by area C+D..Figure 4 Figure 5b.With very inelastic supply and very elastic demand, the burden of the tax on rubberbands will be borne largely by sellers. As Figure 5 shows, consumer surplus does notdecline much, just by area A + B, while producer surplus falls substantially, by area C +D. Compared to part (a), producers bear much more of the burden of the tax, andconsumers bear much less.4. a. The deadweight loss from a tax on heating oil is likely to be greater in the fifth year afterit is imposed rather than the first year. In the first year, the demand for heating oil isrelatively inelastic, as people who own oil heaters are not likely to get rid of them rightaway. But over time they may switch to other energy sources and people buying newheaters for their homes will more likely choose gas or electric, so the tax will have agreater impact on quantity. Thus, the deadweight loss of the tax will get larger over time.b. The tax revenue is likely to be higher in the first year after it is imposed than in the fifthyear. In the first year, demand is more inelastic, so the quantity does not decline asmuch and tax revenue is relatively high. As time passes and more people substitute away from oil, the quantity sold declines, as does tax revenue.5. Because the demand for food is inelastic, a tax on food is a good way to raise revenuebecause it leads to a small deadweight loss; thus taxing food is less inefficient than taxing other things. But it is not a good way to raise revenue from an equity point of view, because poorer people spend a higher proportion of their income on food. The tax would affect them more than it would affect wealthier people.6. a. This tax has such a high rate that it is not likely to raise much revenue. Because of thehigh tax rate, the equilibrium quantity in the market is likely to be at or near zero.b. Senator Moynihan's goal was probably to ban the use of hollow-tipped bullets. In thiscase, the tax could be as effective as an outright ban.7. a. Figure 6 illustrates the market for socks and the effects of the tax. Without a tax, theequilibrium quantity would be Q1, the equilibrium price would be P1, total spending byconsumers equals total revenue for producers, which is P1 x Q1, which equals area B + C + D + E + F, and government revenue is zero. The imposition of a tax places a wedgebetween the price buyers pay, P B, and the price sellers receive, P S, where P B = P S + tax.The quantity sold declines to Q2. Now total spending by consumers is P B x Q2, whichequals area A + B + C + D, total revenue for producers is P S x Q2, which is area C + D,and government tax revenue is Q2 x tax, which is area A + B.b. Unless supply is perfectly elastic or demand is perfectly inelastic, the price received byproducers falls because of the tax. Total receipts for producers fall, because producerslose revenue equal to area B + E + F.Figure 6c. The price paid by consumers rises, unless demand is perfectly elastic or supply isperfectly inelastic. Whether total spending by consumers rises or falls depends on theprice elasticity of demand. If demand is elastic, the percentage decline in quantityexceeds the percentage increase in price, so total spending declines. If demand isinelastic, the percentage decline in quantity is less than the percentage increase in price, so total spending rises. Whether total consumer spending falls or rises, consumer surplus declines because of the increase in price and reduction in quantity.8. Figure 7 illustrates the effects of the $2 subsidy on a good. Without the subsidy, theequilibrium price is P1 and the equilibrium quantity is Q1. With the subsidy, buyers pay price P B, producers receive price P S (where P S = P B + $2), and the quantity sold is Q2. Thefollowing table illustrates the effect of the subsidy on consumer surplus, producer surplus, government revenue, and total surplus. Because total surplus declines by area D + H, the subsidy leads to a deadweight loss in that amount.Figure 79. a. Figure 8 shows the effect of a $10 tax on hotel rooms. The tax revenue is represented byareas A + B, which are equal to ($10)(900) = $9,000. The deadweight loss from the tax is represented by areas C + D, which are equal to (0.5)($10)(100) = $500.Figure 8 Figure 9b. Figure 9 shows the effect of a $20 tax on hotel rooms. The tax revenue is represented byareas A + B, which are equal to ($20)(800) = $16,000. The deadweight loss from the tax is represented by areas C + D, which are equal to (0.5)($20)(200) = $2,000.When the tax is doubled, the tax revenue rises by less than double, while the deadweight loss rises by more than double. The higher tax creates a greater distortion to the market.10. a. Setting quantity supplied equal to quantity demanded gives 2P = 300 –P. Adding P toboth sides of the equation gives 3P = 300. Dividing both sides by 3 gives P = 100.Substituting P = 100 back into either equation for quantity demanded or supplied gives Q = 200.b. Now P is the price received by sellers and P +T is the price paid by buyers. Equatingquantity demanded to quantity supplied gives 2P = 300 − (P+T). Adding P to both sides of the equation gives 3P = 300 –T. Dividing both sides by 3 gives P = 100 –T/3. This is the price received by sellers. The buyers pay a price equal to the price received by sellers plus the tax (P +T = 100 + 2T/3). The quantity sold is now Q = 2P = 200 – 2T/3.c. Because tax revenue is equal to T x Q and Q = 200 – 2T/3, tax revenue equals 200T−2T2/3. Figure 10 (on the next page) shows a graph of this relationship. Tax revenue iszero at T = 0 and at T = 300.Figure 10 Figure 11 d. As Figure 11 shows, the area of the triangle (laid on its side) that represents the deadweight loss is 1/2 × base × height, where the base is the change in the price, which is the size of the tax (T ) and the height is the amount of the decline in quantity (2T /3). So the deadweight loss equals 1/2 × T × 2T /3 = T 2/3. This rises exponentially from 0 (when T = 0) to 30,000 when T = 300, as shown in Figure 12.Figure 12e. A tax of $200 per unit is a bad policy, because tax revenue is declining at that tax level.The government could reduce the tax to $150 per unit, get more tax revenue ($15,000 when the tax is $150 versus $13,333 when the tax is $200), and reduce the deadweight loss (7,500 when the tax is $150 compared to 13,333 when the tax is $200).。
重心三等分三角形面积证明
重心三等分三角形面积证明英文回答：The centroid of a triangle is the point of intersection of the three medians of the triangle. A median of atriangle is a line segment that joins a vertex of the triangle to the midpoint of the opposite side.The centroid of a triangle divides each median into two segments, one of which is twice the length of the other. This means that the centroid is two-thirds of the way from any vertex to the midpoint of the opposite side.The area of a triangle is given by the formula:Area = (1/2) base height.where the base is the length of one side of thetriangle and the height is the perpendicular distance from the opposite vertex to the base.We can use the fact that the centroid is two-thirds of the way from any vertex to the midpoint of the opposite side to find a formula for the area of a triangle in terms of the centroid.Let's consider the triangle ABC with centroid G. Let's also draw the median from A to BC, and let's call the point where the median intersects BC point D.Since G is two-thirds of the way from A to D, we have:AG = 2/3 AD.We can also see that the height of the triangle ABC is equal to GD, since GD is perpendicular to BC and passes through the vertex A.Therefore, we have:Area of triangle ABC = (1/2) BC GD.We can substitute the expression for AG into the expression for the area of the triangle to get:Area of triangle ABC = (1/2) BC (2/3 AD)。
area知识点总结
area知识点总结IntroductionArea is a fundamental concept in mathematics and geometry, used to measure the size of a two-dimensional shape or surface. It helps in understanding the space occupied by an object or a region. This knowledge summary provides a detailed explanation of the concept of area, its formulas, and its applications in various fields.Understanding AreaArea is the amount of space covered by a two-dimensional shape or surface. It is always measured in square units, such as square meters, square centimeters, square inches, etc. The concept of area is crucial in measuring the size of objects, calculating resources required for a project, and understanding spatial relationships.Formulas for Calculating AreaThe area of different shapes can be calculated using specific formulas. Some of the common formulas for calculating the area of different shapes include:- Square: The area of a square is calculated by squaring the length of one of its sides (Area = side x side).- Rectangle: The area of a rectangle is calculated by multiplying its length by its width (Area = length x width).- Triangle: The area of a triangle is calculated using the formula: Area = 1/2 x base x height. - Circle: The area of a circle is calculated using the formula: Area = π x radius^2, where π is a constant approximately equal to 3.14159.- Parallelogram: The area of a parallelogram is calculated by multiplying its base by its height.Applications of AreaThe concept of area has numerous applications in various fields such as mathematics, architecture, construction, agriculture, and physics. Some of the key applications include:- In mathematics, area is used to solve problems related to geometry, trigonometry, and calculus.- In architecture and construction, area is used to calculate the amount of materials required for flooring, tiling, painting, etc.- In agriculture, area is used to measure the size of farmland and calculate the amount of seeds, fertilizers, and pesticides needed for cultivation.- In physics, area is used to calculate the pressure exerted by a force and to understand the concept of flux in electromagnetism.Real-life ExamplesThe concept of area can be better understood through real-life examples. For instance, when planning to carpet a room, the area of the floor needs to be calculated to determine the amount of carpet required. Likewise, when constructing a building, the area of the land and the area of each floor need to be calculated to estimate the total construction cost. In farming, the area of the field is measured to determine the amount of irrigation needed. Therefore, the concept of area is an essential tool for solving practical problems in everyday life.Advanced Topics in AreaIn advanced mathematics, the concept of area extends to the study of calculus, where the calculation of the area under a curve plays a significant role. This is known as finding the integral of a function, which involves calculating the area between the curve and the x-axis. This concept is extensively used in engineering, physics, and other scientific fields.Challenges in Calculating AreaWhile calculating the area of regular shapes like squares, rectangles, and circles is relatively straightforward, calculating the area of irregular shapes can be challenging. In such cases, techniques like decomposing the shape into simpler geometric shapes, using integration in calculus, or applying advanced computational methods may be required.ConclusionIn conclusion, the concept of area is a fundamental aspect of mathematics and plays a crucial role in various fields. Understanding the concept of area, its formulas, and its applications is essential for solving practical problems and gaining insights into spatial relationships. Moreover, the study of advanced topics related to area provides valuable tools for tackling complex problems in mathematics, engineering, and science. Therefore, a comprehensive understanding of the concept of area is indispensable for students, professionals, and enthusiasts in diverse fields.。
四边形对角线互相垂直的面积
四边形对角线互相垂直的面积英文回答：The area of a quadrilateral with perpendicular diagonals can be calculated using different methods, such as the formula for the area of a trapezoid or by dividing the quadrilateral into two triangles and calculating their areas separately.Let's consider a quadrilateral ABCD with perpendicular diagonals AC and BD. We can divide this quadrilateral into two triangles, namely triangle ABC and triangle ACD.To calculate the area of triangle ABC, we can use the formula for the area of a triangle: Area = 1/2 base height. In this case, the base of triangle ABC is AB and the height is the perpendicular distance from point C to line AB. Similarly, to calculate the area of triangle ACD, we use the base AD and the height, which is the perpendicular distance from point B to line AD.Once we have calculated the areas of both triangles, we can add them together to find the total area of the quadrilateral ABCD.For example, let's say AB = 6 units, BC = 4 units, CD = 8 units, and AD = 5 units. The perpendicular distance from point C to line AB is 3 units, and the perpendicular distance from point B to line AD is 2 units.Using the formula for the area of a triangle, we can calculate the area of triangle ABC as 1/2 6 3 = 9 square units. Similarly, the area of triangle ACD is 1/2 5 2 = 5 square units.Adding the areas of both triangles, we get 9 + 5 = 14 square units. Therefore, the area of the quadrilateral ABCD with perpendicular diagonals AC and BD is 14 square units.中文回答：四边形对角线互相垂直的面积可以通过不同的方法计算，比如使用梯形的面积公式或者将四边形分成两个三角形并分别计算它们的面积。
模拟四：Scratch蓝桥杯科学素养模拟测试卷(中级组)
模拟四：Scratch蓝桥杯科学素养模拟测试卷（中级组）姓名：_________班级：_________ [填空题] *1.用手压缩弹簧时可以发现，在弹簧的弹性限度内，（）。
[单选题] *A 手受到的力与压缩距离成正比。
(正确答案)B 手受到的力与压缩距离成反比。
C 手受到的力与压缩距离的平方成正比D 手受到的力与压缩距离的平方成反比2.我们平时食用的土豆是对应植物的（）部分。
[单选题] *A 块茎(正确答案)B 植株C 果实D 花3.人眼看到的苹果是红色的，这是因为（）。
[单选题] *A 苹果表面会吸收红光B 苹果会发出红色的光C 苹果表面反射的光以红色为主(正确答案)D 红色的光打到了苹果上4.影视作品中，人们经常趴在铁轨上听声音判断是否有火车会来。
这个动作主要利用了（）的特性。
[单选题] *A 火车行进时会使得地面震动B 铁轨会和火车共振C 火车的声音在固体中传播比在空气中快，损耗低。
(正确答案)5.生活中我们经常见到铁表面生锈。
生锈实际上是铁表面（）的过程，它属于（）变化。
[单选题] *A 氧化;化学(正确答案)B 磨损;物理C 氧化;物理D 磨损;化学6.使用可控核聚变反应堆发电是人类能源发展的新方向。
在核聚变发电堆中发生的反应与以下（）选项中的发生的反应不同。
[单选题] *A 恒星B 原子弹(正确答案)C 氢弹D 太阳7.在以下选项中，（）距离太阳最远。
[单选题] *A 地球B 水星C 火星D 小行星带(正确答案)8.在同一地点的同一天，（）的温度一般最高。
[单选题] *A 下午四点B 早上九点C 中午12 点D 下午两点(正确答案)9.工业革命以来，人类排放的温室气体急剧增加，成为了全球变暖的主要因素。
以下选项中，不是常见温室气体的是（）。
[单选题] *A二氧化碳（CO2）B 氢气（H2）(正确答案)C水蒸气（H2O）D 甲烷（CH4）10. 在北半球的夏至来临时，以下选项中（）地区的日照时间最长。
IGCSE Math 英文数学词汇知识点整理
1.∅: empty set2.? ∈ A: ∈ for ∈lementi.e. 6 ∈ A3.? ∉ A: NOT an element of Ai.e. 15 ∉ A4.√(y₂−y₁)² + (x₂-x₁)²: Length of a straight line5.A': Complement of set A (not in A)6.A ∩ B: ∩ for i∩tersection7.A ∪ B: ∪ for ∪nion8.a²=b²+c²−2bcCosA: Cosine Rule (side from 2 sides and anincluded angle)9.Acceleration: Gradient on Speed/Time Graphs10.Add any two fractions together: 1) Rewrite fractions asmixed numbers (If there are any)2)Find a common denominator3)Combine the top two lines11.Angles in the same segment are equal.:12.Area of a Rectangle: A x B(Width x Height)13.Area of a Trapezium: ((A+B)/2) * HA = One SideB = Other SideH = Height14.Area of a Triangle: A = 1/2(absinC)15.Area of Circle: Pi x R^2(R = Radius)16.Area of Parellogram: Base x Height17.Area of Sector of a Circle: A= θ/360 x ∏r²18.Area of Triangle: 1/2 x Base x Height19.Area of Triangle: ½ x ab × sin c20.a/SinA = b/SinB = c/SinC: Sine Rule (to find length orangle)21.Average Distance: Speed x Time22.Average Speed: Total Distance travelled / total Time taken23.Average Time: Total Distance travelled / Average Speed24.C ⊂ A: C is a subset (inside) of A 25.Circumference of a Circle: Pi x DiameterOR2 x Pi x Rmon Factor: An integer that divides two (or more) otherintegers evenlymon Multiple: A number into which each number in agiven set may be evenly divided.pound Interest: Interest you get each year is added tothe principal and the next years interest is calculated onincreased amountpound Interest: P(1 + i) ^ nP = Principali = Interest Raten = Number of Years30.CosA=b²+c²-a² / 2bc: Cosine Rule (angle from all 3 sides)31.Cos rule: c² = a² + b² - 2abcosc32.Cos x: adj/hyp33.Directly Proportional: y=kx where k is a constant34.Direct Proportion: y = kx35.Distance Travelled: Area under a Speed-Time graph36.Factor: Any integer that divides into another integer37.Factorize : g² - a²: (g - a)(g +a)38.Factorize : ym + gm + yk +gk =: (y + g)(m + k)39.Factors of a number: all the numbers that divide into it40.Find the HCF of any two numbers using primefactorisation: Prime factorise both numbersThe highest common factor is found by multiplying all the factors which appear in both lists of factors41.Find the LCM of any two numbers using primefactorisation: Prime factorise both numbersThe lowest common multiple is found by multiplying all the factors which appear in either list of factors42.From any point outside the circle, just 2 tangents canbe drawn, and they are of equal length:43.Gradient: change in y / change in x44.Highest Common Factor: The highest number that can bedivided exactly into each of two or more numbers.45.Integers: They are "whole numbers"They can be positive or negativeeg -2, 5, 046.Inversely Proportional: y=k/x where k is a constantIGCSE Math英文数学词汇知识点整理47.Inverse Proportion: y = k/x48.Irrational Number: Number that CANNOT be expressed inthe form p/q where p and q are integers(eg Pi)49.Irrational number: A number that can't be written as afraction50.Length of the arc of a sector of a circle: length = θ/360 x2∏r51.Lower Bound: Estimate less half of the deviatipn52.Lowest Common Multiple: The lowest quantity that is amultiple of two or more given quantities(e.g., 12 is the lowest common multiple of 2, 3, and 4).53.Mean of grouped data: sum of all fX / sum of all f54.n ( A ): Number of elements in set A55.Natural Numbers: The numbers you use to count56.New price to old price: Divide by a factor57.Old price to new price: Multiply by a factor58.Opposite angles of a cyclic quadrilateral add up to180°:59.Percentage Change: Change/Original x 100%60.Percentage Increase/Decrease: Change in Amount ÷Original × 10061.Perimeter of Rectangle: 2 x L + 2 x H(L = Length)(H = Height)62.Prime Factor: A factor that is also a prime numbereg Prime factors of 63 are 3 and 763.Prime Number: Only has two different factors1 is NOT a Primeeg 2,3,5,7,11,13,17,1964.Quadratic Formula: x = (-b ±√(b² - 4ac) )/2a65.r² = (x-a)² + (y-b)²: Formula for a circle66.Rational Number: Number that can be expressed in the formp/q where p and q are whole numbers67.Rational Number: A number that can be written as a fraction68.Ratios of Measurements: A:BA²:B²A³:B³69.Real Number: Number that can be represented on the numberline70.Reverse Percentages: Price after change ÷ 100 ±increase/decrease × 10071.Rules of Indices:72.Simple Interest: P(1 + i x n)P = Principali = Interest Raten = Number of Years73.Simple Interest: You only get interest on the original principleamount74.Simplest Form: Divide either the fraction or the ratio by theHighest Common Factor75.Sin rule: a/sinA = b/sinB = c/sinC76.Sin x: opp/hyp77.Solving for proportionality: 1. Write relation between x andy using k2. Find k using a pair of values & rewrite equation using k3. Use equation to find other values of x & y78.Speed: Gradient on a Distance/Time graph79.Tan x: opp/adj80.The angle at the centre is twice the angle at thecircumference:81.The angle between a tangent and a chord is equal tothe angle in the alternate segment:82.The angle between a tangent and a radius is 90°:83.The angle in a semicircle is 90°:84.Tonne: 1000Kg85.Two methods of finding Prime Factors: factor tree, SHIFTFACT button on calculator86.Upper Bound: Estimate PLUS half of the deviation87.Volume of Cone: 1/3 x Pi x R^2(R = Radius)88.Volume of Cuboid: Width x Length x Height89.Volume of Cylinder: Pi x R^2 x Height(R = Radius)90.Volume of Prism: 1/2 x Base x Height x Length91.Volume of Sphere: 4/3 x Pi x R^3(R = Radius)92.What does the order of operations 'BIDMAS' stand for?:BracketsIndicesDivision or MultiplicationAddition or Subtraction93.x₁+x₂ / 2 , y₁+y₂ / 2: Midpoint of a straight line。
2021AMC8考题及答案
2021AMC8考题及答案1.Which value is the largest among the following?2.Alicia。
Brenda。
XXX votes。
and the XXX?3.What is the value of the given n?4.When 0. is multiplied by 7,928,564.which of the following is the closest product?5.What is the value of the given n?6.If the angles of a triangle are in the。
of。
what is the degree measure of the largest angle in the triangle?7.Let be a 6-digit positive integer。
such as .whose first three digits are the same as its last three digits taken in the same order。
Which of the following numbers must also be a factor of。
8.No paragraph was provided.XXX after school but he doesn't know her house number。
However。
Isabella gives him a hint: her house number has two digits and exactly three of the following four statements are true: it is prime。
even。
divisible by 7.or has one digit of ing this n。