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Middle Years

4.10 Direct and inverse variation

Worksheet
Concept of variation
1

Suppose the constant of variation, k, is positive.

a

If y varies directly with x, describe how y changes as x increases and decreases.

b

If y varies inversely with x, describe how y changes as x increases and decreases.

2

State whether the following equations represent direct or inverse variation:

a
y = 5 x
b
y = \dfrac{5}{x}
3

The period of a pendulum varies directly with the square root of its length. If the length is quadrupled, what happens to the period?

4

State whether the following are examples of direct or inverse variation:

a

The variation relating the distance between two locations on a map and the actual distance between the two locations.

b

The variation relating the number of workers hired to build a house and the time required to build the house.

c

Rhe variation relating the time it takes an ice cube to melt in water and the temperature of the water.

Direct variation
5

In the equation y = 3 x, y varies directly as x.

a

Find the value of y when x = 10.

b

Find the value of y when x = 5.

c

For these two ordered pairs, what is the result when the y value is divided by the x value?

6

Find the equation relating p and q given the table of values:

p36912
q\dfrac{2}{9}\dfrac{1}{18}\dfrac{2}{81}\dfrac{1}{72}
7

Find the equation relating t and s given the table of values:

s1234
t48241612
8

If d = \dfrac{1}{5} s, where d is the approximate distance (in miles) from a storm, and s is the number of seconds between seeing lightning and hearing thunder, describe the relationship between the distance and the number of seconds.

9

If r = \dfrac{d}{t}, where r is the speed when d kilometres in t hours, describe the relationship between the speed and the kilometres.

10

If f = \dfrac{m v^{2}}{r}, where f is the centripetal force of an object of mass m moving along a circle of radius r at velocity v, describe the relationship between the centripetal force and the mass.

11

The mass in grams, M, of a cube of cork varies directly with the cube of the side length in centimetres, x. If a cubic centimetre of cork has a mass of 0.29:

a

Find the constant of variation, k.

b

Express M in terms of x.

c

Find the mass of a cube of cork with a side length of 8 centimetres correct to two decimal places.

12

The surface area, A, of a regular tetrahedron varies directly with the square of its side length, s. A particular tetrahedron with a side length of 2 cm has a surface area of 6.93.

a

Find the constant of variation, k, to two decimal places.

b

Using the rounded value of k, express A in terms of s.

c

Find the surface area of a tetrahedron with a side length of 3 cm to two decimal places.

13

The area, A, of an equilateral triangle varies directly with the square of its side length, s. An equilateral triangle with a side length of 7 cm has an area of 21.22.

a

Find the constant of variation, k, to two decimal places.

b

Express A in terms of s.

c

Find the area of an equilateral triangle with a side length of 2 cm to two decimal places.

14

The number of eggs, n, used in a recipe for a particular cake varies directly with the square of the diameter of the tin, d, for tins with constant depth. 2 eggs are used in a recipe for a tin with a diameter of 17 cm.

a

Find the exact value of the constant of variation, k.

b

How many eggs, n, would be used for a tin with a diameter of 39 cm?

Inverse variation
15

State the meaning of inverse proportion.

16

If r varies inversely with a, write an equation that uses k as the constant of variation.

17

In the equation y = \dfrac{18}{x}, y varies inversely with x. When x = 6, y = 3.

a

Solve for y when x = 2.

b

For these two ordered pairs, what is the result when the y -value is multiplied by the x -value?

18

Suppose that y varies inversely as the square of x, and that y = 0.0023 when x = 0.4.

a

Find the constant of variation, k. Round your answer correct to six decimal places.

b

Write the variation equation for y in terms of x.

19

State whether the following equations are examples of inverse variation:

a

y = \dfrac{7}{x}

b

y = 6 x + 8

c

y = - \dfrac{9}{x}

d

y = \dfrac{8}{x^{2}}

e

y = 2 x^{2} - 7 x - 4

f

y = 3 - x

g

x = 1 + y^{3}

h

x = \dfrac{8}{y^{2}}

i

y = 6 x + 8

j

x y = - 7

k

x = \dfrac{2}{y}

l

x y = 5 x

20

Consider the inverse variation equation y = \dfrac{6}{x}.

a

Complete the following tables:

x\dfrac{1}{4}\dfrac{1}{2}124
y
b

Plot the data from the table of values on a number plane.

21

If y is inversely proportional to x, and y = 20 when x = 10:

a

Find the constant of variation, k.

b

Express y in terms of x.

c

Find the value of y when x = 5.

22

Which of the following tables could represent an inversely proportional relationship between x and y?

A
x1234
y31.510.75
B
x1234
y3618129
C
x15610
y375108300
D
x1234
y4567
23

Find the equation relating n and r given the table of values:

n1234
r5\dfrac{5}{2}\dfrac{5}{3}\dfrac{5}{4}
24

Consider the equation s = \dfrac{375}{t}.

a

State the constant of proportionality.

b

Find the value of s when t = 6.

c

Find the value of s when t = 12.

25

The time, t, taken by a typist to type up a document is inversely proportional to his typing speed, s. That is, the quicker the typing speed, the less time it will take. If it takes a typist 20 minutes to type a particular document, typing at a speed of 61 words per minute:

a

Find the constant of variation k.

b

How long (in minutes) will it take a typist with a typing speed of 30.5 words per minute to type up the document?

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