Variation

- y varies directly as x, and y = 20 when x = 5.

y = kx

20 = k(5)

k = 4

Now replace 4 for k in the original equation

So y = 4x

- y varies directly as x, and y = 9 when x = 27.

y = kx

9 = k(27)

k = 1/3

Now replace 1/3 for k in the original equation

So y = (1/3)x

- y varies directly with x, and y = 20 when x = 16.

y = kx

20 = k(16)

k = 5/4

Now replace 5/4 for k in the original equation

So y = (5/4)x

- y varies directly with x, and y = 32 when x = 20.

y = kx

32 = k(20)

k = 1.6

Now replace 1/3 for k in the original equation

So y = (1.6)x

- y is directly proportional to x, and y = -10 when x = -15.

y = kx

-10 = k(-15)

k = -10/-15

k = 10/15

k = 2/3

Now replace 2/3 for k in the original equation

So y = (2/3)x

- y is directly proportional to x, and y = 300 when x = -60

y = kx

300 = k(-60)

k = 300/-60

k = -5

Now replace -5 for k in the original equation

So y = -5x

- y is directly proportional to x, and y = 17 when x = 17.

y = kx

17 = k(17)

k = 1

Now replace 1 for k in the original equation

So y = 1x

- y varies directly as x, and y = 1.2 when x = 1.6

y = kx

1.2 = k(1.6)

k = (1.2)/(1.6)

k = 3/4

Now replace 3/4 for k in the original equation

So y = (3/4)x

- The distance y traveled at a fixed rate of speed varies directly with the time of travel, x. Write an equation if y = 250 m when x = 25 sec.

y = kx

250 = k(25)

k = 10

Now replace 10 for k in the original equation

So y = 10

- The amount of interest, y, paid on a loan is directly proportional to the amount borrowed, x. Write an equation if y = $75 when x = $500.

y = kx

75 = k(500)

k = 75/500

k = 3/20

Now replace 3/20 for k in the original equation

So y = (3/20)x

- The circumference, y, of a circle varies directly with the diameter, x, of the circle. Write an equation if y = 44 cm when x = 14 cm.

y = kx

44 = k(14)

k = 44/14

k = 22/7

Now replace 22/7 for k in the original equation

So y = (22/7)x

- The amount of money earned on a job is directly proportional to the number of hours worked. If $36 is earned for 8 hours of work, how much is earned for 30 hours of work?

y = kx

36 = k(8)

k = 4.5

Now replace 4.5 for k in the original equation

So y = (4.5)x

Now substitute 30 in for x

y = (4.5)30

y = 135

So $135 is earned for 30 hours of work

- The height that a ball bounces varies directly with the height from which it is dropped. A certain ball bounces 30 cm when dropped from a height of 50 cm. How high will the ball bounce if dropped from a height of 120 cm?

y = kx

30 = k(50)

k = 30/50

k = 3/5

Now replace 3/5 for k in the original equation

So y = (3/5)x

Now substitute 120 in for x

y = (3/5)120

y = 72

So the ball bounces 72 cm when dropped from 120 cm

- The amount that a spring stretches is directly proportional to the weight of the object attached to it. If a spring is stretched 10cm by a weight of 8 kg, how much will it be stretched by a weight of 3 kg?

y = kx

10 = k(8)

k = 5/4

Now replace 5/4 for k in the original equation

So y = (5/4)x

Now substitute 3 in for x

y = (5/4)3

y = 15/4

So the spring stretches 3.75 cm when a weight of 3 kg is attached.

- The number of calories in a container of milk is directly proportional to the amount of milk in the container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15-ounce glass of milk.

y = kx

160 = k(8)

k = 20

Now replace 20 for k in the original equation

So y = 20x

Now substitute 15 in for x

y = 20(15)

y = 300

So there are 300 calories in a 15 ounce of milk

- The number of kilograms of water in a person’s body varies directly as the person’s mass. A person with a mass of 90 kg contains 60 kg of water. How many kilograms of water are in a person with a mass of 50 kg?

y = kx

90 = k(60)

k = 90/60

k = 3/2

Now replace 3/2 for k in the original equation

So y = (3/2)x

y = 50

50 = (3/2)x

100/3 = x

x = 33 1/3

So a person with a mass of 50 kg would have 33 1/3 kilograms of water

- On a certain map, 25 km are represented by 2 cm. If two cities are 7 cm apart on the map, what is the actual distance between them?

y = kx

25 = k(2)

k = 25/2

k = 12.5

Now replace 12.5 for k in the original equation

So y = (12.5)x

y = 12.5(7)

y = 87.5

So the 2 cities are 87.5 km apart.

- The amount of fertilizer needed for a lawn varies directly with the area of the lawn. If 4 pounds of fertilizer are needed for 500 square feet of lawn, how much is needed for Dr. Quagmire’s lawn, which is rectangular in shape and measures 25 feet by 50 feet?
- y varies inversely as x, and y = 25 when x = 3.20. y varies inversely as x, and y = 7 when x = 12.21. y inversely proportional to x, and y = 3.5 when x = 8.
22. y inversely proportional to x, and y = 0.4 when x = 0.9.

23. The time, T, it takes to travel a certain distance caries inversely as the speed, s. Write an equation it t = 10 h when s = 80 km/h.

24. The length, l, of a rectangle with a constant area varies inversely as the width, w. Write an equation if l = 7.2 when w = 5.0.

25. The time, t, required to do a certain job is inversely proportional to the number of people, n, working. Write an equation if t = 15h when n = 6.

26. y varies inversely as x, and y = -320 when x = -5.

27. y is inversely proportional to x, and y = 0.125 when x = -100.

28. y varies inversely as x, and y = -9 when x = 28.

29. y is inversely proportional to x, and y = 2.5 when x = 0.4.

30. The number of chairs, y, on a ski lift is inversely proportional to the distance, x, between them. Write an equation if y = 40 when x = 30m.

31. The force, F, need to lift an object with a crowbar varies inversely with the length, L, of the crowbar. Write an equation if F = 200kg when L = 1.4m.

32. The frequency, f, of a sound wave is inversely proportional to the wavelength, L. Write an equation if f = 420Hz when L = 0.8m.

- For rectangles with the same area, the length varies inversely as the width. One rectangle has a length of 12cm and a width of 5 cm. Find the length of another rectangle with the same area whose width is 4 cm.
- The current in an electrical Circuit varies inversely as the amount of resistance in the circuit. The current is 10 amps when the resistance is 24 ohms. Find the current when the resistance is 30 ohms.
- The cost per person to rent a mountain cabin is inversely proportional to the number of people who share the rent. If the cost is $36 per person when 5 people share, what is the cost per person when 8 people share?
- The volume of a gas varies inversely as the pressure. A helium-filled balloon has a volume of 21 m
^{3 }at sea level, where the pressure is 1 atmosphere. The balloon rises to an altitude where the pressure is 0.7 atmospheres. What is its volume? - The number of chairs on a ski lift is inversely proportional to the distance between them. A lift has 70 chairs when they are spaced 24 m apart. If 80 evenly-spaced chairs are used on the lift, how much space will be left between them?
- For piano wires under the same tension, the number of vibrations per second (frequency) of each wire is inversely proportional to the length of the wire. A wire 0.75 m long vibrates 480 times per second. How long is a wire that vibrates 300 times per second?
- The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If the trip takes 6 h at 900 km/h, how long would it take at 800 kmh?
- The distance, d, that a free-falling body falls varies directly as the square of the time, t, that it falls. If d = 36 m when t = 3 sec,

Find the value of k.

Find d when t = 5 sec.

- The amount of material, M, needed to cover a ball is directly proportional to the square of the radius, r. If M = 60 cm
^{2}when r = 2 cm,

Find the value of k.

Find M when r = 2 cm.

- The price, p, of a pizza varies directly as the square of its radius, r. If p = $6.00 when r = 10 cm,

Find the value of k.

Find p when r = 15 cm.

- The brightness of illumination, I, of an object varies inversely as the square of its distance, d, from the source of illumination. If I = 18 Iuxes when d = 4 m,

Find the value of k.

Find I when d = 3 m.

- The time, t, needed to fill the gas tank of a car varies inversely as the square of the diameter, d, of the hose. If t = 5 min when d = 3 cm,

Find the value of k.

Find t when d = 2 cm.

- The electrical resistance, R, of a wire of a certain length is inversely proportional to the square of its diameter, d. If R = 10 ohms when d= 0.6 mm,

Find the value of k.

Find r when d = 3 mm.

- The price, p, of a diamond is directly proportional to the square of its weight, w. If p = $2000 when w = 1 carat,

Find the value of k.

Find p when w = 0.7 carat.

- V varies jointly as B and h.
- t varies directly as W and inversely as n.
- P varies directly as the square of V and inversely as R.
- h varies directly as W and inversely as the square of r.
- E varies jointly as m and the square of v.
- I varies jointly as A and H and inversely as T.
- The mass, m, of a cement block varies jointly as the length, l, width, w, and thickness, t, of the block.
- The volume, v, of a gas varies directly as the temperature, T, and inversely as the pressure, P.
- The collision impact, I, of an automobile varies jointly as the mass, m, and the square of the speed, s.
- The intensity of a sound, i, varies directly as the amplitude, a of the sound source, and inversely as the square of the distance, d, from the source.
- The safe load, s, for a beam, varies jointly as the breadth, b, and the square of the depth, d, and inversely as the length, l, between supports.
- The gravitational force, g, between two objects varies jointly as the mass of the first, m
_{1}, and the mass of the second, m_{2}, and inversely as the square of the distance, d, between them.