Functions and Their Graphs

SECTION
3.6

Modeling Functions Using Variation

SKILLS OBJECTIVES


■��	Develop mathematical models using direct variation.

■��	Develop mathematical models using inverse variation.

■��	Develop mathematical models using combined variation.

■��	Develop mathematical models using joint variation.

CONCEPTUAL OBJECTIVES


■��	Understand the difference between direct variation and inverse variation.

■��	Understand the difference between combined variation and joint variation.

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In this section we discuss mathematical models for different applications. Two quantities in the real world often vary with respect to one another. Sometimes, they vary directly. For example, the more money we make, the more total dollars of federal income tax we expect to pay. Sometimes, quantities vary inversely. For example, when interest rates on mortgages decrease, we expect the number of homes purchased to increase because a buyer can afford “more house” with the same mortgage payment when rates are lower. In this section we discuss quantities varying directly, inversely, and jointly.

Direct Variation

When one quantity is a constant multiple of another quantity, we say that the quantities are directly proportional to one another.

DIRECT VARIATION

Let

and

represent two quantities. The following are equivalent statements:


■��	, where is a nonzero constant.

■��	varies directly with .

■��	is directly proportional to .

The constant

is called the constant of variation or the constant of proportionality.

In 2005, the national average cost of residential electricity was

(cents per kilowatt-hour). For example, if a residence used 3400 kWh, then the bill would be

, and if a residence used 2500 kWh, then the bill would be

EXAMPLE�1�

Finding the Constant of Variation

In the United States, the cost of electricity is directly proportional to the number of

(kWh) used. If a household in Tennessee on average used

per month and had an average monthly electric bill of

, find a mathematical model that gives the cost of electricity in Tennessee in terms of the number of

used.

Solution

■	YOUR TURN�

Find a mathematical model that describes the cost of electricity in California if the cost is directly proportional to the number of kWh used and a residence that consumes 4000 kWh is billed

Not all variation we see in nature is direct variation. Isometric growth, where the various parts of an organism grow in direct proportion to each other, is rare in living organisms. If organisms grew isometrically, young children would look just like adults, only smaller. In contrast, most organisms grow nonisometrically; the various parts of organisms do not increase in size in a one-to-one ratio. The relative proportions of a human body change dramatically as the human grows. Children have proportionately larger heads and shorter legs than adults. Allometric growth is the pattern of growth whereby different parts of the body grow at different rates with respect to each other. Some human body characteristics vary directly, and others can be mathematically modeled by direct variation with powers.

DIRECT VARIATION WITH POWERS

Let

and

represent two quantities. The following are equivalent statements:


■��	, where is a nonzero constant.

■��	varies directly with the power of .

■��	is directly proportional to the power of .

One example of direct variation with powers is height and weight of humans. Weight (in pounds) is directly proportional to the cube of height (feet).

�EXAMPLE�2�

Direct Variation with Powers

The following is a personal ad:

Single professional male (6 ft/194 lbs) seeks single professional female for long-term relationship. Must be athletic, smart, like the movies and dogs, and have height and weight similarly proportioned to mine.

Find a mathematical equation that describes the height and weight of the male who wrote the ad. How much would a

woman weigh who has the same proportionality as the male?

Solution

Write the direct variation (cube) model for height versus weight.
Substitute the given data and into .
Solve for .
�
Let .

A woman

tall with the same height and weight proportionality as the male would weigh

■	YOUR TURN�

A brother and sister both have weight (pounds) that varies as the cube of height (feet) and they share the same proportionality constant. The sister is 6 feet tall and weighs 170 pounds. Her brother is 6 feet 4 inches. How much does he weigh?

Inverse Variation

Two fundamental topics covered in economics are supply and demand. Supply is the quantity that producers are willing to sell at a given price. For example, an artist may be willing to paint and sell 5 portraits if each sells for

, but that same artist may be willing to sell 100 portraits if each sells for

. Demand is the quantity of a good that consumers are not only willing to purchase but also have the capacity to buy at a given price. For example, consumers may purchase 1 billion Big Macs from McDonald's every year, but perhaps only 1 million filet mignons are sold at Outback. There may be 1 billion people who want to buy the filet mignon but don't have the financial means to do so. Economists study the equilibrium between supply and demand.

Demand can be modeled with an inverse variation of price: when the price increases, demand decreases, and vice versa.

INVERSE VARIATION

Let

and

represent two quantities. The following are equivalent statements:


■��	, where is a nonzero constant.

■��	varies inversely with .

■��	is inversely proportional to .

The constant

is called the constant of variation or the constant of proportionality.

�EXAMPLE�3�

Inverse Variation

The number of potential buyers of a house decreases as the price of the house increases (see graph on the right). If the number of potential buyers of a house in a particular city is inversely proportional to the price of the house, find a mathematical equation that describes the demand for houses as it relates to price. How many potential buyers will there be for a

million house?

Solution

Write the inverse variation model.
Label the variables and constant.
�
Select any point that lies on the curve.
Substitute the given data and into .
Solve for .
�
Let .

■	YOUR TURN�

In New York City, the number of potential buyers in the housing market is inversely proportional to the price of a house. If there are 12,500 potential buyers for a

million condominium, how many potential buyers are there for a

million condominium?

Two quantities can vary inversely with the

power of

and

are related by the equation

, then we say that

varies inversely with the

power of

, or

is inversely proportional to the

power of

Joint Variation and Combined Variation

We now discuss combinations of variations. When one quantity is proportional to the product of two or more other quantities, the variation is called joint variation. When direct variation and inverse variation occur at the same time, the variation is called combined variation.

An example of a joint variation is simple interest (Section 1.2), which is defined as

where


■��	is the interest in dollars

■��	is the principal (initial) dollars

■��	is the interest rate (expressed in decimal form)

■��	is time in years

The interest earned is proportional to the product of three quantities (principal, interest rate, and time). Note that if the interest rate increases, then the interest earned also increases. Similarly, if either the initial investment (principal) or the time the money is invested increases, then the interest earned also increases.

An example of combined variation is the combined gas law in chemistry,

where


■��	is pressure

■��	is temperature (kelvins)

■��	is volume

■��	is a gas constant

This relation states that the pressure of a gas is directly proportional to the temperature and inversely proportional to the volume containing the gas. For example, as the temperature increases, the pressure increases, but when the volume decreases, pressure increases.

As an example, the gas in the headspace of a soda bottle has a fixed volume. Therefore, as temperature increases, the pressure increases. Compare the different pressures of opening a twist-off cap on a bottle of soda that is cold versus one that is hot. The hot one feels as though it “releases more pressure.”

�EXAMPLE�4�

Combined Variation

The gas in the headspace of a soda bottle has a volume of

, pressure of 2 atm (atmospheres), and a temperature of

(standard room temperature of

). If the soda bottle is stored in a refrigerator, the temperature drops to approximately

. What is the pressure of the gas in the headspace once the bottle is chilled?

Solution

Write the combined gas law.
Let , , and .
Solve for .
Let , , and in .
Since we used the same physical units for both the chilled and room-temperature soda bottles, the pressure is in atmospheres.

�

SECTION
3.6

SUMMARY

Direct, inverse, joint, and combined variation can be used to model the relationship between two quantities. For two quantities

and

, we say that


■��	is directly proportional to if .

■��	is inversely proportional to if .

Joint variation occurs when one quantity is directly proportional to two or more quantities. Combined variation occurs when one quantity is directly proportional to one or more quantities and inversely proportional to one or more other quantities.

SECTION
3.6

EXERCISES

SKILLS

In Exercises 1-16, write an equation that describes each variation. Use

as the constant of variation.

1.��	varies directly with .

2.��	varies directly with .

3.��	varies directly with .

4.��	varies directly with .

5.��	varies directly with .

6.��	varies directly with .

7.��	varies inversely with .

8.��	varies inversely with .

9.��	varies directly with and inversely with .

10.��

varies directly with

and inversely with

11.��

varies directly with both

and

12.��

varies directly with both

and

13.��

varies inversely with both

and

14.��

varies inversely with both

and

15.��

is directly proportional to the square root of

16.��

is inversely proportional to the cube of

In Exercises 17-36, write an equation that describes each variation.

17.��

is directly proportional to

when

18.��

is directly proportional to

when

19.��

is directly proportional to both

and

when

and

20.��

is directly proportional to both

and

when

and

21.��

varies directly with the square of

when

22.��

varies directly with the cube of

when

23.��

varies directly with both

and

when

and

24.��

is directly proportional to both

and the square of

when

and

25.��

varies inversely with

when

26.��

varies inversely with the square of

when

27.��

varies inversely with both

and

when

and

28.��

varies inversely with both

and

when

and

29.��

varies inversely with

when

30.��

varies inversely with the square of

when

31.��

varies inversely with the square of

when

32.��

varies inversely with both

and the square root of

when

and

33.��

varies directly with

and inversely with

when

and

34.��

varies directly with

and inversely with

when

and

35.��

varies directly with both

and

and inversely with the square of

when

, and

36.��

varies directly with the square root of

and inversely with the square of

when

and

APPLICATIONS

37.��

Wages.

Jason and Valerie both work at Panera Bread and have the following paycheck information for a certain week. Find an equation that shows their wages

varying directly with the number of hours worked

Employee	Hours Worked	Wages
Jason
Valerie

38.��

Sales Tax.

The sales tax in Orange and Seminole counties in Florida differs by only 0.5%. A new resident knows this but doesn't know which of the counties has the higher tax. The resident lives near the border of the counties and is in the market for a new plasma television and wants to purchase it in the county with the lower tax. If the tax on a pair of

sneakers is

in Orange County and the tax on a

T-shirt is

in Seminole County, write two equations: one for each county that describes the tax

, which is directly proportional to the purchase price

For Exercises 39 and 40, refer to the following:

The ratio of the speed of an object to the speed of sound determines the Mach number. Aircraft traveling at a subsonic speed (less than the speed of sound) have a Mach number less than 1. In other words, the speed of an aircraft is directly proportional to its Mach number. Aircraft traveling at a supersonic speed (greater than the speed of sound) have a Mach number greater than 1. The speed of sound at sea level is approximately 760 miles per hour.

39.��

Military.

The U.S. Navy Blue Angels fly F-18 Hornets that are capable of Mach 1.7. How fast can F-18 Hornets fly at sea level?

40.��

Military.

The U.S. Air Force's newest fighter aircraft is the F-35, which is capable of Mach 1.9. How fast can an F-35 fly at sea level?

Exercises 41 and 42 are examples of the golden ratio, or phi, a proportionality constant that appears in nature. The numerical approximate value of phi is 1.618. From www.goldenratio.net.

41.��

Human Anatomy.

The length of your forearm

(wrist to elbow) is directly proportional to the length of your hand

(length from wrist to tip of middle finger). Write the equation that describes this relationship if the length of your forearm is 11 inches and the length of your hand is 6.8 inches.

42.��

Human Anatomy.

Each section of your index finger, from the tip to the base of the wrist, is larger than the preceding one by about the golden (Fibonacci) ratio. Find an equation that represents the ratio of each section of your finger related to the previous one if one section is eight units long and the next section is five units long.

For Exercises 43 and 44, refer to the following:

Hooke's law in physics states that if a spring at rest (equilibrium position) has a weight attached to it, then the distance the spring stretches is directly proportional to the force (weight), according to the formula:

where

is the force in Newtons (N),

is the distance stretched in meters (m), and

is the spring constant (N/m).

43.��

Physics.

A force of

will stretch the spring 10 centimeters. How far will a force of

stretch the spring?

44.��

Physics.

A force of

will stretch the spring 10 centimeters. How much force is required to stretch the spring 18 centimeters?

45.��

Business.

A cell phone company develops a pay-as-you-go cell phone plan in which the monthly cost varies directly as the number of minutes used. If the company charges

in a month when 236 minutes are used, what should the company charge for a month in which 500 minutes are used?

46.��

Economics.

Demand for a product varies inversely with the price per unit of the product. Demand for the product is 10,000 units when the price is

per unit. Find the demand for the product (to the nearest hundred units) when the price is

47.��

Sales.

Levi's makes jeans in a variety of price ranges for juniors. The Flare 519 jeans sell for about

, whereas the 646 Vintage Flare jeans sell for

. The demand for Levi's jeans is inversely proportional to the price. If 300,000 pairs of the 519 jeans were bought, approximately how many of the Vintage Flare jeans were bought?

48.��

Sales.

Levi's makes jeans in a variety of price ranges for men. The Silver Tab Baggy jeans sell for about

, whereas the Offender jeans sell for about

. The demand for Levi's jeans is inversely proportional to the price. If 400,000 pairs of the Silver Tab Baggy jeans were bought, approximately how many of the Offender jeans were bought?

For Exercises 49 and 50, refer to the following:

In physics, the inverse square law states that any physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity. In particular, the intensity of light radiating from a point source is inversely proportional to the square of the distance from the source. Below is a table of average distances from the Sun:

Planet	Distance to the Sun
Mercury
Earth
Mars

49.��

Solar Radiation.

The solar radiation on the Earth is approximately 1400 watts per square meter

. How much solar radiation is there on Mars? Round to the nearest hundred watts per square meter.

50.��

Solar Radiation.

The solar radiation on the Earth is approximately 1400 watts per square meter. How much solar radiation is there on Mercury? Round to the nearest hundred watts per square meter.

51.��

Investments.

Marilyn receives a

bonus from her company and decides to put the money toward a new car that she will need in two years. Simple interest is directly proportional to the principal and the time invested. She compares two different banks’ rates on money market accounts. If she goes with Bank of America, she will earn

in interest, but if she goes with the Navy Federal Credit Union, she will earn

. What is the interest rate on money market accounts at both banks?

52.��

Investments.

Connie and Alvaro sell their house and buy a fixer-upper house. They made

on the sale of their previous home. They know it will take 6 months before the general contractor will start their renovation, and they want to take advantage of a 6-month CD that pays simple interest. What is the rate of the 6-month CD if they will make

in interest?

53.��

Chemistry.

A gas contained in a 4 milliliter container at a temperature of

has a pressure of 1 atmosphere. If the temperature decreases to

, what is the resulting pressure?

54.��

Chemistry.

A gas contained in a 4 milliliter container at a temperature of

has a pressure of 1 atmosphere. If the container changes to a volume of 3 millileters, what is the resulting pressure?

CATCH THE MISTAKE

In Exercises 55 and 56, explain the mistake that is made.

55.��

varies directly with

and indirectly with

. When

and

, then

. Find an equation that describes this variation.

Solution:

Write the variation equation.
Let , , and .
Solve for .
Substitute into .

This is incorrect. What mistake was made?

56.��

varies directly with

and the square of

. When

and

, then

. Find an equation that describes this variation.

Solution:

Write the variation equation.
Let , , and .
Solve for .
Substitute into .

This is incorrect. What mistake was made?

CONCEPTUAL

In Exercises 57 and 58, determine whether each statement is true or false.

57.��

The area of a triangle is directly proportional to both the base and the height of the triangle (joint variation).

58.��

Average speed is directly proportional to both distance and time (joint variation).

In Exercises 59 and 60, match the variation with the graph.

(a)��

(b)��

59.��

Inverse variation

60.��

Direct variation

CHALLENGE

Exercises 61 and 62 involve the theory governing laser propagation through the Earth's atmosphere.

The three parameters that help classify the strength of optical turbulence are:


■��	, index of refraction structure parameter

■��	, wave number of the laser, which is inversely proportional to the wavelength of the laser:

■��	, propagation distance

The variance of the irradiance of a laser

is directly proportional to

, and

61.��

When

, and

, the variance of irradiance for a plane wave

is 7.1. Find the equation that describes this variation.

62.��

When

, and

, the variance of irradiance for a spherical wave

is 2.3. Find the equation that describes this variation.

TECHNOLOGY

For Exercises 63-66, refer to the following:

Data from 1995 to 2006 for oil prices in dollars per barrel, the U.S. Dow Jones Utilities Stock Index, New Privately Owned Housing, and 5-year Treasury Constant Maturity Rate are given in the table. (Data are from Forecast Center's Historical Economic and Market Home Page at www.neatideas.com/djutil.htm.)

Use the calculator

commands to enter the table with

as the oil price,

as the utilities stock index,

as number of housing units, and

as the 5-year maturity rate.

January of Each Year	Oil Price, $ per Barrel	U.S. Dow Jones Utilities Stock Index	New, Privately Owned Housing Units	5-Year Treasury Constant Maturity Rate
1995	17.99	193.12	1407	7.76
1996	18.88	230.85	1467	5.36
1997	25.17	232.53	1355	6.33
1998	16.71	263.29	1525	5.42
1999	12.47	302.80	1748	4.60
2000	27.18	315.14	1636	6.58
2001	29.58	372.32	1600	4.86
2002	19.67	285.71	1698	4.34
2003	32.94	207.75	1853	3.05
2004	32.27	271.94	1911	3.12
2005	46.84	343.46	2137	3.71
2006	65.51	413.84	2265	4.35

63.��

An increase in oil price in dollars per barrel will drive the U.S. Dow Jones Utilities Stock Index to soar.

(a)��

Use the calculator commands

, and

to model the data using the least squares regression. Find the equation of the least-squares regression line using

as the oil price in dollars per barrel.

(b)��

If the U.S. Dow Jones Utilities Stock Index varies directly as the oil price in dollars per barrel, then use the calculator commands

, and

to model the data using the power function. Find the variation constant and equation of variation using

as the oil price in dollars per barrel.

(c)��

Use the equations you found in (a) and (b) to predict the stock index when the oil price hits

per barrel in September 2006. Which answer is closer to the actual stock index of 417? Round all answers to the nearest whole number.

64.��

An increase in oil price in dollars per barrel will affect the interest rates across the board—in particular, the 5-year Treasury constant maturity rate.

(a)��

Use the calculator commands

, and

to model the data using the least-squares regression. Find the equation of the least-squares regression line using

as the oil price in dollars per barrel.

(b)��

If the 5-year Treasury constant maturity rate varies inversely as the oil price in dollars per barrel, then use the calculator commands

, and

to model the data using the power function. Find the variation constant and equation of variation using

as the oil price in dollars per barrel.

(c)��

Use the equations you found in (a) and (b) to predict the maturity rate when the oil price hits

per barrel in September 2006. Which answer is closer to the actual maturity rate at

? Round all answers to two decimal places.

65.��

An increase in interest rates—in particular, the 5-year Treasury constant maturity rate—will affect the number of new, privately owned housing units.

(a)��

Use the calculator commands

, and

to model the data using the least-squares regression. Find the equation of the least-squares regression line using

as the 5-year rate.

(b)��

If the number of new privately owned housing units varies inversely as the 5-year Treasury constant maturity rate, then use the calculator commands

, and

to model the data using the power function. Find the variation constant and equation of variation using

as the 5-year rate.

(c)��

Use the equations you found in (a) and (b) to predict the number of housing units when the maturity rate is

in September 2006. Which answer is closer to the actual number of new, privately owned housing units of

? Round all answers to the nearest unit.

66.��

An increase in the number of new, privately owned housing units will affect the U.S. Dow Jones Utilities Stock Index.

(a)��

Use the calculator commands

, and

to model the data using the least-squares regression. Find the equation of the least-squares regression line using

as the number of housing units.

(b)��

If the U.S. Dow Jones Utilities Stock Index varies directly as the number of new, privately owned housing units, then use the calculator commands

, and

to model the data using the power function. Find the variation constant and equation of variation using

as the number of housing units.

(c)��

Use the equations you found in (a) and (b) to predict the utilities stock index if there are 1861 new, privately owned housing units in September 2006. Which answer is closer to the actual stock index of 417? Round all answers to the nearest whole number.

For Exercises 67 and 68, refer to the following:

Data for retail gasoline price in dollars per gallon for the period March 2000 to March 2008 are given in the following table. (Data are from Energy Information Administration, Official Energy Statistics from the U.S. government at http://tonto.eia.doe.gov/oog/info/gdu/gaspump.html.) Use the calculator

command to enter the table below with

as the year (

for year 2000) and

as the gasoline price in dollars per gallon.

March of each year	2000	2001	2002	2003	2004	2005	2006	2007	2008
Retail gasoline price $ per gallon	1.517	1.409	1.249	1.693	1.736	2.079	2.425	2.563	3.244

67.��

(a)��

Use the calculator commands

to model the data using the least-squares regression. Find the equation of the least-squares regression line using

as the year (

for year 2000) and

as the gasoline price in dollars per gallon. Round all answers to three decimal places.

(b)��

Use the equation to predict the gasoline price in March 2006. Round all answers to three decimal places. Is the answer close to the actual price?

(c)��

Use the equation to predict the gasoline price in March 2009. Round all answers to three decimal places.

68.��

(a)��

Use the calculator commands

to model the data using the power function. Find the variation constant and equation of variation using

as the year (

for year 2000) and

as the gasoline price in dollars per gallon. Round all answers to three decimal places.

(b)��

Use the equation to predict the gasoline price in March 2006. Round all answers to three decimal places. Is the answer close to the actual price?

(c)��

Use the equation to predict the gasoline price in March 2009. Round all answers to three decimal places.