中学用英文编写的数学讲义：比、比例及变数法

Solution
Volume of the sphere : Volume of the cylinder
指数律
r1 r2
3 3

27 8
a3 a ( )3 3 b b 3 x k
x3 k
r1 3 27 r2 8

r1 3 r2 2
2
(C) Proportion
A proportion is an equality of two ratios. Note: If a : b = c : d, where b 0 and d 0 , then (i) (ii) ad = bc
r
球体表面面积(S)公式: S = 4r 2
Solution Let S m2 denotes the surface area of the spherical container and $P denotes the cost. Since , we have When r = 5, Substituting S = 100 and P = 2000 into P = kS, .
Variation
Direct Variation In words : y varies (directly) as x or y is (directly) proportional to x In symbols : Equation : O y
Graph of y = kx .
x (k>0)
Chapter8 Ratio, Proportion and Variation(比、比例及变数法)
The Syllabus Whole Syllabus Including direct, inverse, joint and partial variations(正变、反变、联变及部分变法). Tailored Syllabus Including direct, inverse, joint and partial variations. Excluding algebraic manipulation of continued proportion(连续比例).

8
Exercise 8 - 1(Class Work)
Matching 1. 2. 3. 4. 5. 6. 7. 8. If z varies directly as x2 , en If z varies inversely as x2, then If z varies directly as x2 and inversely as y, then If z varies inversely as x2 and directly as y, then If z varies jointly as x2 and y, then If z is partly constant and partly varies directly as x2, then If z is partly varies directly as x2 and partly varies directly as y, then If z is partly varies inversely as x2 and partly varies directly as y, then A. B. C. D. E. F. G. H.
(ii)
Inverse Variation In words : y varies inversely as x y is inversely proportional to x In symbols : O
y Graph of y =
k . x
or
x (k>0)
Equation : 图象中 y 值随 x 值的增加而减少， 而图象不会接触 x 轴及 y 轴 。 6
ab . a c2
2
k-method 很重要，必须熟练。
Example 7 If a : b = c : d, where Solution Let
, show that ,where k is a non-zero constant.
.
Then a = bk and c = dk
4
(D)
(i)
z k1 x 2 k 2 y z kx2 y k z 1 k2 y x2 kx2 z y z kx2
z k1 k 2 x 2
z
Equation : (iii)Joint Variation In words : z varies jointly as x and y In symbols : z xy Equation : z = kxy, k 0 . Example 11 If z varies jointly as x and inversely as y, and x = 1, y = 2, and z = 3, express z in terms of x and y . Solution In symbols : z
Examples 2 The heights of John and Mary are 125cm and 150cm. The ratio of the height of John to the height of Mary is 125 : 150. 5 i.e. 5 : 6 or 6
1
Examples 3 Figure(a) shows a sphere of radius r 1 and Figure(b) shows a cylinder of base radius r 2 and height r 2. If the ratio of the volume of the sphere to the volume of the cylinder is 9 : 2, find r 1 : r 2.
Continued proportion
If
then each ratio =
,
where l1, l2, l3,… are constants such that
Example 5 a b c 5a 6b c If , find x. 2 3 4 x Solution 可改用 k-method 做 E.g. 5. 即 设 a = 2k, b = 3k 及 c = 4k。 .................(i) 得k
x y
Equation : z =
kx ,k 0. y
Find k : Sub. x = 1, y = 2 and z = 3
z
6x y
7
(iv)Partial Variation If z partly varies (directly) as x and partly varies (directly) as y, then , where k1 and k2 are non-zero constants. z Graph of z = a + kx .
Example 8
图象是一条通过原心 (origin），斜率(slope) 是 k 的 直线。
If y varies directly as x2 and y = 2, when x = 1, find the equation connecting x and y. Solution In symbols : Equation : .
100 km/h 2 = 50 km/h = 50000 m/h (b) The travelling rate = 50000m/h
(a) The travelling rate =
(B)
Ratio
A ratio is a comparison between two numbers or two quantities of the Same kind and ratio has No unit. Note: (i) If k 0 ,a : b = ka : kb = (ii) If a : b = m : n and b : c = n : k, then a : b : c = m : n : k
(A)Rate
A rate is a comparison between two quantities of Different kinds and rate has unit. Example 1 A car travels 100km in 2 hours. The rate of travelling of the car is 50km/h. Find the speed of the car in (a) metre per hour, (b) metre per second.(Correct the answer to 3 significant figures) Solution
Find k : Sub. x = -1 and y = 2

5
Example 9 The cost of painting a spherical container of radius r cm in varies directly as the surface area. If r = 5, the cost is $2000. Find the cost when r = 8.
5(2k ) 6(3k ) 4k x x=?
.................(ii) Compare (i) and (ii), we have 3
x 4
k-method When a : b : c = 2 : 3 : 4, we have Let, then a = 2k, b = 3k and c = 4k. Example 6 If a : b : c = 2 : 3 : 4, Find Solution Let a = 2k, b = 3k and c = 4k, k 0 .
Example 10 If y varies inversely as Solution In symbols : , and y = -2, when x = 4, find the relation between x and y.
Equation : Find k : Sub. x = 4 and y = -2
a kb c kd , k is any real number. b d
Example 4 If a : b = 2 : 1 and a : c = 3 : 5, find a : b : c. Solution a:b=2:1=6:3 a : c = 3 : 5 = 6 : 10 a : b : c = 6 : 3 : 10
If z is partly constant and partly varies (directly) as x, then, z = a + kx, where a is a constant and k is a non-zero constant.
a O x (a >0 and k>0)
图象是一条 y 轴截距( y - intercept)是 a，斜率(slope) 是 k 的直线。 Example 12 Given that y partly varies as x and partly varies as x2. When x = -1, y = 1; when x = 1, y = 3. Express y in terms of x. Solution Let Solution Find k1 and k2 : Sub. x = -1, y = 1; x = 1, y = 3 注意题目中用来分辨联变(Joint variation) 及 部分变(Partial variation)的重要字眼，即 ' partly varies as ...... partly varies as ' 。 , where k1 and k2 are non-zero constants.