Direct Variation and 8.1 Inverse Direct Variation
Direct Variation – y varies directly as x or is directly proportional.
y = kx n , k ≠ 0 and n is a positive integer ≥ 1
(does not have to be linear)
y
y
OR 1 = 2
(this ratio gives the constant of variation, k)
x1 x2
Example 1: The radius, r (in inches), of a hailstone varies directly with the time, t (in
seconds), that the hailstone is in the high cloud. After a hailstone has been in a certain
cloud for 10 seconds, the radius of the hailstone is 0.5 inch. Write a linear model that
gives the radius of the hailstone in terms of time t and then determine the constant of
variation.
1. The area, A (in square inches), of a rectangle varies directly with its width,
w (in inches). When the width is 4 inches, the area is 12 square inches. Write
an equation that relates A and w.
Y varies directly as x. Find the value as indicated for 2-6 below.
2. If y = 12 when x = 4, find y when x = 12.
3. If y = 16 when x = -4, find y when x = 8.
4. If y = -12 when x = -3, find y when x = -8.
5. If y = 5 when x = 25, find y when x = 30.
1
6. If y = 3 when x = 10, find x when y = .
2
7. If y varies directly as the cube of x, and x = 0.2 when y = 16, find y when x
= 0.5.
8. The stretch S in a spring balance varies directly as the applied weight w. If
S = 5 inches when w = 15 lb. find the weight needed for a stretch of 7.5
inches.
9. When you swim underwater, the pressure in your ears varies directly with
the depth at which you swim. At 10 feet, the pressure is about 4.3 pounds per
square inch (psi).
a) Write the particular equation expressing pressure in terms of depth.
b) Predict the pressure at 50 feet.
c) It is unsafe for amateur divers to swim where the pressure is more
than 65 psi. How deep can an amateur diver safely swim?
Inverse Variation – y varies inversely as x or is inversely proportional.
k
k ≠ 0 and n is a positive integer ≥ 1
xn
(does not have to be linear)
y
y
OR 2 = 1
(this ratio gives the constant of variation, k)
x1 x2
y=
Example 1: The number of songs, n, that can be stored on an MP3 player varies
inversely with the average size of a song, s (in megabytes). A certain MP3 player can
store 2500 songs when the average size of a song is 4 megabytes (MB). Write a linear
model that gives the number of songs that will fit on the MP3 player in terms of size s
and then determine the constant of variation.
1. The area, A (in square millimeters), of a computer chip varies inversely
with the number of chips that can be obtained from a silicon wafer, c. When
the number of chips is 448, the area is 58 square millimeters. Write an
equation that relates A and c.
Y varies inversely as x. Find the value as indicated for 2-6 below.
2. If y = 12 when x = 4, find y when x = 12.
3. If y = 16 when x = -4, find y when x = 8.
4. If y = -12 when x = -3, find y when x = -8.
5. If y = 5 when x = 25, find y when x = 30.
1
6. If y = 3 when x = 10, find x when y = .
2