8.9 Use Direct and Inverse Variation

Learning objectives

Learning objectives

• Solve direct variation problems

• Solve inverse variation problems

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.

Solve Direct Variation Problems

Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation

Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.

In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.

Example 8.86

We’ll list the steps below.

Now we’ll solve a few applications of direct variation.

Example 8.87

Solution

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In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems.

Example 8.88

Solution

ⓑ $\begin{array}{cccl} & & & \mathrm{Find}\,g\,\mathrm{when}\,m=1000. \\ \mathrm{Write the equation that relates}\,g\,\mathrm{and}\,m. & & & g=0.031m \\ \mathrm{Substitute the given value for}\,m. & & & g=0.031(1000) \\ \mathrm{Simplify.} & & & g=31 \\ & & & \mathrm{Eunice's car would use 31 gallons of gas if she drove it 1,000 miles.} \\ \end{array}$

Notice that in this example, the units on the constant of variation are gallons/mile. In everyday life, we usually talk about miles/gallon.

In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is $y=k\,{x}^{2}$. We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for k.

Example 8.89

Solution

ⓑ $\begin{array}{cccl} & & & \,\mathrm{Find}\,L\,\mathrm{when}\,c=8. \\ \mathrm{Write the equation that relates}\,L\,\mathrm{and}\,c. & & & \,L=4.6875{c}^{2} \\ \mathrm{Substitute the given value for}\,c. & & & \,L=4.6875{(8)}^{2} \\ \mathrm{Simplify.} & & & \,L=300 \\ & & & \,\begin{array}{l} \mathrm{A beam with diagonal 8" could support} \\ \mathrm{a maximum load of 300 pounds.} \\ \end{array} \\ \end{array}$

Solve Inverse Variation Problems

Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is $y=\frac{k}{x}$.

The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is $\frac{1}{x}$.

We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.

Example 8.90

Solution

Example 8.91

Solution

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Example 8.92

Solution

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Section 8.9 Exercises

Section 8.9 Exercises

Solve Direct Variation Problems

In the following exercises, solve.

Solve Inverse Variation Problems

In the following exercises, solve.

Write an inverse variation equation to solve the following problems.

Mixed Practice

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

1. If $y$ varies directly as $x$ and $y=3,\,\mathrm{when}\,x=10.$ find the equation that relates x and y.

2. If $y$ varies directly as $x$ and $y=12\,\mathrm{when}\,x=4$ find the equation that relates x and y.

3. The number of calories, c, burned varies directly with the amount of time, t, spent exercising. Arnold burned 312 calories in 65 minutes exercising.

1. ⓐ Write the equation that relates c and t.

2. ⓑ How many calories would he burn if he exercises for 90 minutes?

1. The distance a moving body travels, d, varies directly with time, t, it moves. A train travels 100 miles in 2 hours

ⓐ Write the equation that relates d and t. ⓑ How many miles would it travel in 5 hours?

2. The distance that Brad travels varies directly with the time spent traveling. Brad travelled 660 miles in 12 hours,

1. ⓐ Write the equation that relates the number of miles travelled to the time.

2. ⓑ How many miles could Brad travel in 4 hours?

3. The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons.

1. ⓐ Write the equation that relates the weight to the volume.

2. ⓑ If a liquid has volume 13 gallons, what is its weight?

1. The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds.

1. ⓐ Write the equation that relates the distance to the time.

2. ⓑ How far will an object fall in 4 seconds?

2. The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.

1. ⓐ Write the equation that relates the area to the radius.

2. ⓑ What is the area of a pizza with a radius of 9 inches?

3. If $p$ varies inversely with $q$ and $p=30$ when $q=12$ find the equation that relates $p$ and $q.$

1. If $y$ varies inversely with $x$ and $y=8$ when $x=2$ find the equation that relates $x$ and $y$.

2. A car’s value varies inversely with its age. Elena bought a two-year-old car for $20,000.

ⓐ Write the equation of variation. ⓑ What will be the value of Elena’s car when it is 5 years old?

3. The time required to empty a pool varies inversely as the rate of pumping. It took Lucy 2.5 hours to empty her pool using a pump that was rated at 400 gpm (gallons per minute).

1. ⓐ Write the equation of variation.

2. ⓑ How long will it take her to empty the pool using a pump rated at 500 gpm?

1. The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees.

1. ⓐ Write the equation of variation.

2. ⓑ How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?

2. The force needed to break a board varies inversely with its length. Richard uses 24 pounds of pressure to break a 2-foot long board.

1. ⓐ Write the equation of variation.

2. ⓑ How many pounds of pressure is needed to break a 5-foot long board?

3. If $y$ varies directly as $x$ and $y=14,\,\mathrm{when}\,x=3$, find the equation that relates $x\,\mathrm{and}\,y$.

1. If $v$ varies directly as $w$ and $v=24,\,\mathrm{when}\,w=8$, find the equation that relates $v\,\mathrm{and}\,w.$

2. If $p$ varies directly as $q$ and $p=9.6,\,\mathrm{when}\,q=3$, find the equation that relates $p\,\mathrm{and}\,q.$

3. If $a$ varies directly as $b$ and $a=6,\,\mathrm{when}\,b=\frac{1}{3}$, find the equation that relates $a\,\mathrm{and}\,b.$

1. The amount of money Sally earns, P, varies directly with the number, n, of necklaces she sells. When Sally sells 15 necklaces she earns $150.

1. ⓐ Write the equation that relates P and n.

2. ⓑ How much money would she earn if she sold 4 necklaces?

2. Terri needs to make some pies for a fundraiser. The number of apples, a, varies directly with number of pies, p. It takes nine apples to make two pies.

1. ⓐ Write the equation that relates a and p.

2. ⓑ How many apples would Terri need for six pies?

3. The price of gas that Jesse purchased varies directly to how many gallons he purchased. He purchased 10 gallons of gas for $39.80.

1. ⓐ Write the equation that relates the price to the number of gallons.

2. ⓑ How much will it cost Jesse for 15 gallons of gas?

1. The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.

1. ⓐ Write the equation that relates the mass to the volume.

2. ⓑ What is the volume of this liquid if its mass is 128 kilograms?

2. The distance an object falls varies directly to the square of the time it falls. A ball falls 45 feet in 3 seconds.

1. ⓐ Write the equation that relates the distance to the time.

2. ⓑ How far will the ball fall in 7 seconds?

3. The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.

1. ⓐ Write the equation that relates the area to the radius.

2. ⓑ What is the area of a personal pizza with a radius 4 inches?

1. If $y$ varies inversely with $x$ and $y=5$ when $x=4$ find the equation that relates $x$ and $y.$

2. If $v$ varies inversely with $w$ and $v=6$ when $w=\frac{1}{2}$ find the equation that relates $v$ and $w.$

3. The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds and gets 33 mpg on the highway.

1. ⓐ Write the equation that relates the mpg to the car’s weight.

2. ⓑ What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?

1. The time required to empty a tank varies inversely as the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute),

1. ⓐ Write the equation that relates the number of hours to the pump rate.

2. ⓑ How long would it take Janet to pump her basement if she used a pump rated at 400 gpm?

2. On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second.

1. ⓐ Write the equation that relates the string length to its frequency.

2. ⓑ What is the frequency of a 10-inch string?

3. The number of tickets for a sports fundraiser varies inversely to the price of each ticket. Brianna can buy 25 tickets at $5each.

1. ⓐ Write the equation that relates the number of tickets to the price of each ticket.

2. ⓑ How many tickets could Brianna buy if the price of each ticket was $2.50?

1. If $y$ varies directly as $x$ and $y=5,\,\mathrm{when}\,x=3.$, find the equation that relates $x\,\mathrm{and}\,y$.

2. If $p$ varies inversely with $q$ and $p=5$ when $q=6$, find the equation that relates $p$ and $q.$

3. If $p$ varies directly as $q$ and $p=10,\,\mathrm{when}\,q=2.$ find the equation that relates $p\,\mathrm{and}\,q$.

1. The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot long board, how many pounds of pressure would he need to use to break a 6 foot long board?

2. The length a spring stretches varies directly with a weight placed at the end of the spring. When Meredith placed a 6-pound cantaloupe on a hanging scale, the spring stretched 2 inches. How far would the spring stretch if the cantaloupe weighed 9 pounds?

3. The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that weighs 5,500 pounds? Round to the nearest tenth.

1. The distance an object falls varies directly to the square of the time it falls. If an object falls 52.8 feet in 4 seconds, how far will it fall in 9 seconds?

2. Ride Service It costs $35 for a ride from the city center to the airport, 14 miles away.

1. ⓐ Write the equation that relates the cost, c, with the number of miles, m.

2. ⓑ What would it cost to travel 22 miles with this service?

3. In your own words, explain the difference between direct variation and inverse variation.