 | SECTION 3.6 | Modeling Functions Using 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:
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, where 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 
.
| EXAMPLE1 | 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
| 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:
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, where 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).
| EXAMPLE2 | 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. |
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Substitute the given data |
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Solve for |
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Let |
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and 
into 
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A woman tall with the same height and weight proportionality as the male would weigh 
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| 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:
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, where The constant is called the constant of variation or the constant of proportionality.
| EXAMPLE3 | 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 |
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million house?
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Solution
Write the inverse variation model. |
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Label the variables and constant. |
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Select any point that lies on the curve. |
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Substitute the given data |
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Solve for |
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Let |
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and 
into 
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| 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?Two quantities can vary inversely with the 
power of .
If 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
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is the interest in dollars
is the principal (initial) dollars
is the interest rate (expressed in decimal form)
is time in yearsThe 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
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is temperature (kelvins)
is volumeThis 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.”
| EXAMPLE4 | 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. |
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Let |
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Solve for |
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Let |
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Since we used the same physical units for both the chilled and room-temperature soda bottles, the pressure is in atmospheres. |
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, 
, and 
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, and 
in 
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Direct, inverse, joint, and combined variation can be used to model the relationship between two quantities. For two quantities
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 |
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In Exercises 1-16, write an equation that describes each variation. Use as the constant of variation.
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| 2. |
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varies directly with | 3. |
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varies directly with 
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varies directly with 
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varies directly with 
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varies inversely with 
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and inversely with 
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| 11. |
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varies directly with both 
and | 12. |
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varies directly with both 
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varies inversely with both | 14. |
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| 15. |
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| 16. |
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In Exercises 17-36, write an equation that describes each variation.
| 17. |
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when 
.| 18. |
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when 
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and 
. 
when 
and 
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and 
when 
and 
.| 21. |
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when 
.| 22. |
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when | 23. |
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when 
and 
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is directly proportional to both 
when 
and 
.| 25. |
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when 
.| 26. |
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when 
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when 
and 
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when 
and 
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when 
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when 
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when 
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when 
and 
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when 
and 
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when 
and 
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and 
and inversely with the square of 
when 
, 
, and 
.| 36. |
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when 
and 
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| APPLICATIONS |
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| 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
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| 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 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. | 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
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| 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.
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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 |
| 45. | 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 |
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Mercury | |
Earth |
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Mars |
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| 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 |
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In Exercises 55 and 56, explain the mistake that is made.
| 55. |
Solution:
This is incorrect. What mistake was made? |
, then 
. Find an equation that describes this variation.
into 
| 56. |
Solution:
This is incorrect. What mistake was made? |
. Find an equation that describes this variation.
| CONCEPTUAL |
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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.
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| 59. | Inverse variation |
| 60. | Direct variation |
| CHALLENGE |
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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:
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, index of refraction structure parameter
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 |
is 2.3. Find the equation that describes this variation.
| TECHNOLOGY |
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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.
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, and 
to model the data using the least squares regression. Find the equation of the least-squares regression line using 
, and 
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.
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? 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.
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? 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.
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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. |
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to model the data using the least-squares regression. Find the equation of the least-squares regression line using | 68. |
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