7.3 Ratio, Proportion, and Variation

Part 2: Direct and Indirect Variation

Direct Variation

• The table shows the rates a carpet cleaning services charges.

• Notice that if you divide the cost by the number of rooms, you will get $49.99 each time.

• This relationship is an example of direct variation.– y varies directly as x such that y = kx, where k is a

numerical value called the constant of variation.

# of Rooms

Cost

1 $49.99

2 $99.98

3 $149.97

4 $199.96

5 $249.95

Solving a Direct Variation Problem

• Suppose y varies directly as x, and y = 50 when x = 20. Find y when x = 14.

Solving a Direct Variation Problem

• Hooke’s law for an elastic spring states that the distance a spring stretches is directly proportional to the force applied. If a force of 150 lbs stretches a certain spring 8 cm, how much will a force of 400 lbs stretch the spring?

If y varies directly as x, and y = 30 when x = 8, find y when x = 4.

Direct Variation as a Power

• In some cases one quantity will vary directly as a power of another.– y varies directly as the nth power of x such

that y = kxn.

• Example: The area of a circle is A = πr2. Here, π is the constant of variation and the area varies directly as the square of the radius.

Solving a Direct Variation Problem

• The distance a body falls from rest varies directly as the square of the time it falls (we are disregarding air resistance). If a skydiver falls 64 feet in 2 seconds, how far will she fall in 8 seconds?

If a varies directly as b2, and a = 48 when b = 4, find a when b = 7.

Inverse Variation

• In direct variation, as x increases so does y and as x decreases so does y.

• Another type of variation is inverse variation.

• y varies inversely as x such that

• Also, y varies inversely as the nth power of x such that

.k

yx

.n

ky

x

Solving an Inverse Variation Problem

• The weight of an object above Earth varies inversely as the square of its distance from the center of Earth. A space vehicle in an elliptical orbit has a maximum distance from the center of Earth (apogee) of 6700 mi. Its minimum distance from the center of Earth (perigee) is 4090 mi. If an astronaut in the vehicle weighs 47 lbs at its apogee, what does she weigh at the perigee?

apogee

perigee

For a constant area, the length of a rectangle varies inversely as the width. The length of a rectangle is 27 ft when the width is 10 ft. Find the length of a rectangle with the same area if the width is 18 ft.

Joint and Combined Variation

• If one variable varies as the product of several other variables (perhaps raised to powers), the first variable is said to vary jointly as the others.– ie: y = kxz, y= kx2z, etc.

• There are situations that involve combinations of direct and inverse variation. These are known as combined variation problems.

Solving a Joint Variation Problem

• The strength of a rectangular beam varies jointly as its width and the square of its depth. If the strength of a beam 2 in. wide by 10 in. deep is 1000 lbs per in.2, what is the strength of a beam 4 in. wide and 8 in. deep?

The weight of a trout varies jointly as its length and the square of its girth. One trout weighs 10.5 lbs and measures 26 in. long with an 18-in. girth. Find the weight of a trout that is 22 in. long with a 15-in. girth.

Solving a Combined Variation Problem

• BMI varies directly as an individual’s weight in pounds and inversely as the square of the individual’s height in inches, rounding to the nearest whole number (a “good” BMI is between 19 and 25). A person who weighs 118 lbs and is 64 in. tall has a BMI of 20. Find the BMI of a person who weighs 165 lbs with a height of 70 in.