Chapter 1

The per capita personal income in the United States from 1998 to 2005 can be approximated by the linear equation \(y=944.7 t+19,898, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998\. Use the model to estimate the year in which the per capita personal income was \(\$ 32,000\).

The demand equation for a product is \(p=60-0.0004 x\) where \(p\) is the price per unit and \(x\) is the number of units sold. The total revenue \(R\) for selling \(x\) units is given by \(R=x p\) How many units must be sold to produce a revenue of \(\$ 220,000 ?\)

Comparing Investment Returns You have \(\$ 10,000\) in an account earning simple interest that is linked to the prime rate. The prime rate drops for the last quarter of the year, so your rate drops by \(1 \frac{1}{2} \%\) for the same period. Your total annual interest is \(\$ 1112.50\). What is your interest rate for the first three quarters and for the last quarter?

Annual Sales The annual sales \(S\) (in billions of dollars) of Microsoft Corporation from 1996 to 2006 can be approximated by the linear equation \(S=3.54 t-13.1, \quad 6 \leq t \leq 16\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. Use the model to estimate the year in which Microsoft's annual sales were about \(\$ 20,000,000,000\).

The demand equation for a product is \(p=50-0.0005 x\) where \(p\) is the price per unit and \(x\) is the number of units sold. The total revenue \(R\) for selling \(x\) units is given by \(R=x p\) How many units must be sold to produce a revenue of \(\$ 250,000 ?\)

Assume that air resistance is negligible, which implies that the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) is a reasonable model Wind Resistance At the same time a skydiver jumps from an airplane 13,000 feet above the ground, a steel ball is dropped from the plane. Because of air resistance, it takes the skydiver 67 seconds to freefall to a height of 3000 feet where the parachute opens. The steel ball has relatively no air resistance, so its height can be modeled by the position equation. How much faster does the ball reach a height of 3000 feet than the skydiver?

Production Limit \(\ln\) Exercises 75 and 76, use the following information. Variable costs depend on the number of units produced. Fixed costs are the same regardless of how many units are produced. Find the greatest number of units the company can produce each month. The company has fixed monthly costs of \(\$ 15,000\) and variable monthly costs of \(\$ 8.75\) per unit. The company has \(\$ 90,000\) available each month to cover costs.

In Exercises 75 and 76 , use the following information. The relationship between the length of an adult's femur (thigh bone) and the height of the adult can be approximated by the linear equations \(y=0.432 x-10.44\) \(y=0.449 x-12.15\) where \(y\) is the length of the femur in inches and \(x\) is the height of the adult in inches (see figure). An anthropologist discovers a femur belonging to an adult human female. The bone is 15 inches long. Estimate the height of the female.

A company weighs each 16-ounce bag of flour it produces. After production, any bag that does not weigh within \(0.4\) ounce of 16 ounces cannot be sold. Solve the equation \(|x-16|=0.4\) to find the least and greatest acceptable weights of a 16 -ounce bag of flour.

Production Limit \(\ln\) Exercises 75 and 76, use the following information. Variable costs depend on the number of units produced. Fixed costs are the same regardless of how many units are produced. Find the greatest number of units the company can produce each month. The company has fixed monthly costs of \(\$ 10,000\) and variable monthly costs of \(\$ 9.30\) per unit. The company has \(\$ 85,000\) available each month to cover costs.

Use the following information. The relationship between the length of an adult's femur (thigh bone) and the height of the adult can be approximated by the linear equations \(y=0.432 x-10.44\) \(y=0.449 x-12.15\) where \(y\) is the length of the femur in inches and \(x\) is the height of the adult in inches (see figure). From the foot bones of an adult human male, an anthropologist estimates that the male was 65 inches tall. A few feet away from the site where the foot bones were discovered, the anthropologist discovers an adult male femur that is 17 inches long. Is it possible that the leg and foot bones came from the same person? Explain.

A company weighs each 80 -ounce bag of sugar it produces. After production, any bag that does not weigh within \(1.2\) ounces of 80 ounces cannot be sold. Solve the equation \(|x-80|=1.2\) to find the least and greatest acceptable weights of an 80 -ounce bag of sugar.

Geometry The hypotenuse of an isosceles right triangle is 6 centimeters long. How long are the legs? (An isosceles right triangle is one whose two legs are of equal length.)

Length of a Tank The diameter of a cylindrical propane gas tank is 4 feet (see figure). The total volume of the tank is \(603.2\) cubic feet. Find the length of the tank.

With only the cold water valve open, it takes 8 minutes to fill the tub of a washing machine. With both the hot and cold water valves open, it takes 5 minutes. The time it takes for the tub to fill with only the hot water valve open can be modeled by the equation \(\frac{1}{8}+\frac{1}{t}=\frac{1}{5}\) where \(t\) is the time (in minutes) for the tub to fill. How long does it take for the tub of the washing machine to fill with only the hot water valve open?

Geometry An equilateral triangle has a height of 3 feet. How long are each of its legs? (Hint: Use the height of the triangle to partition the triangle into two right triangles of the same size.)

Water Depth \(\mathrm{A}\) trough is 12 feet long, 3 feet deep, and 3 feet wide (see figure). Find the depth of the water when the trough contains 70 gallons of water. (1 gallon \(\approx 0.13368\) cubic foot.)

Use the following information. From 1998 to 2005, the annual credit \(y\) (in billions of dollars) extended to consumers in the United States (other than real estate loans) can be approximated by the equation \(y=129.51 t+320.5, \quad 8 \leq t \leq 15\) where \(t\) is the year, with \(t=8\) corresponding to 1998. Use the model to predict the year in which the credit extended to consumers will be about \(\$ 2.9\) trillion.

The cost of renting a midsize car from Company A is \(\$ 279\) per week with no extra charge for mileage. The cost of renting a similar car from Company B is \(\$ 199\) per week, plus 32 cents for each mile driven. How many miles must you drive in a week to make the rental fee for Company \(\mathrm{B}\) greater than that for Company \(\mathrm{A}\) ?

You and a friend volunteer to paint a small house as a community service project. Working alone, you can paint the house in 15 hours. Your friend can paint the house in 18 hours working alone. How long will it take both of you, working together, to paint the house?

In Exercises 79 and 80 , use the following information. From 1997 to 2006, the federal minimum wage was \(\$ 5.15\) per hour. Adjusting for inflation, the federal minimum wage's value in 1996 dollars during these years can be approximated by the linear equation \(y=-0.112 t+5.83, \quad 7 \leq t \leq 16\) where \(t\) is the year, with \(t=7\) corresponding to 1997. In which year was the value of the federal minimum wage about \(\$ 4.60\) in 1996 dollars?

Your department sends its copying to a photocopy center. The photocopy center bills your department \(\$ 0.08\) per page. You are considering buying a departmental copier for \(\$ 2500 .\) With your own copier the cost per page would be \(\$ 0.025 .\) The expected life of the copier is 4 years. How many copies must you make in the four-year period to justify purchasing the copier?

You and a friend volunteer to paint a large house as a community service project. Working alone, you can paint the house in 28 hours. Your friend can paint the house in 25 hours working alone. How long will it take both of you, working together, to paint the house?

Mixture A farmer mixes gasoline and oil to make 2 gallons of mixture for his two-cycle chain saw engine. This mixture is 32 parts gasoline and 1 part two- cycle oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil?

For \(\$ 1500\) to grow to more than \(\$ 1890\) in 3 years, what must the simple interest rate be?

Use the following information. The sum of the angles of a triangle is \(180^{\circ}\). Also, if two angles of a triangle are equal, the lengths of the sides opposite the angles are equal. Depth of a Whale Shark A research ship is tracking the movements of a whale shark that is 700 meters from the ship. The angle formed by the ocean surface and a line from the ship to the whale shark is \(45^{\circ}\). How deep is the whale shark?

New York City Marathon In Exercises 81 and 82, the length of the New York City Marathon course is 26 miles, 385 yards. Find the average speed of the record holding runner. (Note that 1 mile \(=5280\) feet \(=\) 1760 yards.) Men's record time: 2 hours, \(7 \frac{3}{4}\) minutes

For \$2000 to grow to more than \(\$ 2500\) in 2 years, what must the simple interest rate be?

College Costs The average yearly cost \(C\) of attending a private college full time for the academic years \(1999 / 2000\) to \(2004 / 2005\) in the United States can be approximated by the model \(C=45.6 t^{2}+15,737, \quad 10 \leq t \leq 15\) where \(t=10\) corresponds to the \(1999 / 2000\) academic year (see figure). Use the model to predict the year in which the average cost of attending a private college full time is about \$30,000. (Source: U.S. National Center for Education Statistics)

A person enrolls in a diet program that guarantees a loss of at least \(1 \frac{1}{2}\) pounds per week. The person's weight at the beginning of the program is 180 pounds. Find the maximum number of weeks before the person attains a weight of 130 pounds.

Total Revenue The demand equation for a product is \(p=36-0.0003 x\), where \(p\) is the price per unit and \(x\) is the number of units sold. The total revenue \(R\) for selling \(x\) units is given by \(R=x p=x(36-0.0003 x) .\) How many units must be sold to produce a revenue of \(\$ 1,080,000 ?\)

In Exercises 83-100, solve for the indicated variable. Area of a Triangle Solve for \(h\) in \(A=\frac{1}{2} b h\).

You accept a new job with a starting salary of \(\$ 28,800\). You are told that you will receive an annual raise of at least \(\$ 1500\). What is the maximum number of years you must work before your annual salary will be \(\$ 40,000\) ?

Total Revenue The demand equation for a product is \(p=40-0.0005 x\), where \(p\) is the price per unit and \(x\) is the number of units sold. The total revenue \(R\) for selling \(x\) units is given by \(R=x p=x(40-0.0005 x)\) How many units must be sold to produce a revenue of \(\$ 800,000 ?\)

Solve for the indicated variable. Perimeter of a Rectangle Solve for \(l\) in \(P=2 l+2 w\).

An overnight delivery service will not accept any package whose combined length and girth (perimeter of a cross section) exceeds 132 inches. Suppose that you are sending a rectangular package that has square cross sections. If the length of the package is 68 inches, what is the maximum width of the sides of its square cross sections?

Production Cost A company determines that the average monthly cost \(C\) (in dollars) of raw materials for manufacturing a product line can be modeled by \(C=35.65 t^{2}+7205, \quad t \geq 0\) where \(t\) is the year, with \(t=0\) corresponding to 2000 . Use the model to estimate the year in which the average monthly cost reaches \(\$ 12,000\).

An overnight delivery service will not accept any package whose combined length and girth (perimeter of a cross section) exceeds 126 inches. Suppose that you are sending a rectangular package that has square cross sections. If the length of the package is 66 inches, what is the maximum width of the sides of its square cross sections?

Monthly Cost A company determines that the average monthly cost \(C\) (in dollars) for staffing temporary positions can be modeled by \(C=135.47 t^{2}+13,702, \quad t \geq 0\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . Use the model to predict the year in which the average monthly cost is about \(\$ 25,000\).

Solve for the indicated variable. Ideal Gas Law Solve for \(T\) in \(P V=n R T\).

Analysis The revenue \(R\) for selling \(x\) units of a product is \(R=139.95 x\) The cost \(C\) of producing \(x\) units is \(C=97 x+850\) In order to obtain a profit, the revenue must be greater than the cost. (a) Complete the table. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline R & & & & & & \\ \hline C & & & & & & \\ \hline \end{array} $$ (b) For what values of \(x\) will this product return a profit?

Solve for the indicated variable. Volume of a Right Circular Cylinder Solve for \(h\) in \(V=\pi r^{2} h\)

The revenue \(R\) for selling \(x\) units of a product is \(R=25.95 x .\) The cost \(C\) of producing \(x\) units is $$ C=13.95 x+125,000 $$ In order to obtain a profit, the revenue must be greater than the cost. For what values of \(x\) will this product return a profit?

U.S. Population The resident population \(P\) (in thousands) of the United States from 1900 to 2000 can be approximated by the model \(P=1951.00 t^{2}+97,551, \quad 0 \leq t \leq 10\) where \(t\) represents the year, with \(t=0\) corresponding to \(1900, t=1\) corresponding to 1910 , and so on (see figure). Assume this model continues to be valid. In what year will the resident population of the United States reach \(330,000,000 ?\) (Source: U.S. Census Bureau)

Solve for the indicated variable. Kinetic Energy Solve for \(m\) in \(E=\frac{1}{2} m v^{2}\).

A utility company has a fleet of vans. The annual operating cost \(C\) per van is $$ C=0.32 m+2500 $$ where \(m\) is the number of miles traveled by a van in a year. What number of miles will yield an annual operating cost that is less than \(\$ 12,000 ?\)

Solve for the indicated variable. Markup Solve for \(C\) in \(S=C+R C\)

A doughnut shop sells a dozen doughnuts for \(\$ 3.95 .\) Beyond the fixed costs (rent, utilities, and insurance) of \(\$ 165\) per day, it costs \(\$ 1.45\) for enough materials (flour, sugar, and so on) and labor to produce a dozen doughnuts. The daily profit from doughnut sales varies between \(\$ 100\) and \(\$ 400\). Between what numbers of doughnuts (in dozens) do the daily sales vary?

MAKE A DECISION The enrollment \(E\) in an early childhood development program for a school district from 1995 to 2008 can be approximated by the model \(E=1.678 t^{2}+1025,5 \leq t \leq 18\), where \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use the model to approximate the year in which the early childhood enrollment reached 1450 children. Can you use the model to estimate early childhood enrollment for the year 1980 ? Explain.

Solve for the indicated variable. Discount Solve for \(L\) in \(S=L-R L\).