4.12 Direct and inverse variation
Suppose the constant of variation, k, is positive.
If y varies directly with x, describe how y changes as x increases and decreases.
If y varies inversely with x, describe how y changes as x increases and decreases.
State whether the following equations represent direct or inverse variation:
The period of a pendulum varies directly with the square root of its length. If the length is quadrupled, what happens to the period?
State whether the following are examples of direct or inverse variation:
The variation relating the distance between two locations on a map and the actual distance between the two locations.
The variation relating the number of workers hired to build a house and the time required to build the house.
Rhe variation relating the time it takes an ice cube to melt in water and the temperature of the water.
In the equation y = 3 x, y varies directly as x.
Find the value of y when x = 10.
Find the value of y when x = 5.
For these two ordered pairs, what is the result when the y value is divided by the x value?
Find the equation relating p and q given the table of values:
| p | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| q | \dfrac{2}{9} | \dfrac{1}{18} | \dfrac{2}{81} | \dfrac{1}{72} |
Find the equation relating t and s given the table of values:
| s | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| t | 48 | 24 | 16 | 12 |
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.
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.
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.
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:
Find the constant of variation, k.
Express M in terms of x.
Find the mass of a cube of cork with a side length of 8 centimetres correct to two decimal places.
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.
Find the constant of variation, k, to two decimal places.
Using the rounded value of k, express A in terms of s.
Find the surface area of a tetrahedron with a side length of 3 cm to two decimal places.
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.
Find the constant of variation, k, to two decimal places.
Express A in terms of s.
Find the area of an equilateral triangle with a side length of 2 cm to two decimal places.
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.
Find the exact value of the constant of variation, k.
How many eggs, n, would be used for a tin with a diameter of 39 cm?
State the meaning of inverse proportion.
If r varies inversely with a, write an equation that uses k as the constant of variation.
In the equation y = \dfrac{18}{x}, y varies inversely with x. When x = 6, y = 3.
Solve for y when x = 2.
For these two ordered pairs, what is the result when the y -value is multiplied by the x -value?
Suppose that y varies inversely as the square of x, and that y = 0.0023 when x = 0.4.
Find the constant of variation, k. Round your answer correct to six decimal places.
Write the variation equation for y in terms of x.
State whether the following equations are examples of inverse variation:
y = \dfrac{7}{x}
y = 6 x + 8
y = - \dfrac{9}{x}
y = \dfrac{8}{x^{2}}
y = 2 x^{2} - 7 x - 4
y = 3 - x
x = 1 + y^{3}
x = \dfrac{8}{y^{2}}
y = 6 x + 8
x y = - 7
x = \dfrac{2}{y}
x y = 5 x
Consider the inverse variation equation y = \dfrac{6}{x}.
Complete the following tables:
| x | \dfrac{1}{4} | \dfrac{1}{2} | 1 | 2 | 4 |
|---|---|---|---|---|---|
| y |
Plot the data from the table of values on a number plane.
If y is inversely proportional to x, and y = 20 when x = 10:
Find the constant of variation, k.
Express y in terms of x.
Find the value of y when x = 5.
Which of the following tables could represent an inversely proportional relationship between x and y?
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 3 | 1.5 | 1 | 0.75 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 36 | 18 | 12 | 9 |
| x | 1 | 5 | 6 | 10 |
|---|---|---|---|---|
| y | 3 | 75 | 108 | 300 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 4 | 5 | 6 | 7 |
Find the equation relating n and r given the table of values:
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| r | 5 | \dfrac{5}{2} | \dfrac{5}{3} | \dfrac{5}{4} |
Consider the equation s = \dfrac{375}{t}.
State the constant of proportionality.
Find the value of s when t = 6.
Find the value of s when t = 12.
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:
Find the constant of variation k.
How long (in minutes) will it take a typist with a typing speed of 30.5 words per minute to type up the document?