Chapter 1

Distance to a Star Find the distance (in miles) to a star that is 50 light years (distance traveled by light in 1 year) away. (Light travels at 186,000 miles per second.)

Use a calculator to solve the equation. (Round your solution to three decimal places.) \((x+5.62)^{2}+10.83=(x+7)^{2}\)

The revenue \(R\) and cost \(C\) for a product are given by \(R=x(50-0.0002 x)\) and \(C=12 x+150,000\), where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold (see figure). (a) How many units must be sold to obtain a profit of at least \(\$ 1,650,000 ?\) (b) The demand equation for the product is \(p=50-0.0002 x\) where \(p\) is the price per unit. What prices will produce a profit of at least \(\$ 1,650,000 ?\) (c) As the number of units increases, the revenue eventually decreases. After this point, at what number of units is the revenue approximately equal to the cost? How should this affect the company's decision about the level of production?

Solve the inequality. Then graph the solution set on the real number line. \(|9-2 x|-2<-1\)

Compound Interest A deposit of \(\$ 3000\) reaches a balance of \(\$ 4296.16\) after 6 years. The interest on the account is compounded monthly. What is the annual interest rate for this investment?

A small commuter airline flies to three cities whose locations form the vertices of a right triangle (see figure). The total flight distance (from City A to City B to City \(C\) and back to City \(A\) ) is 1400 miles. It is 600 miles between the two cities that are farthest apart. Find the other two distances between cities.

Geometry A one-story building is 14 feet longer than it is wide (see figure). The building has 1632 square feet of floor space. What are the dimensions of the building?

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(\frac{2}{7.398}-\frac{4.405}{x}=\frac{1}{x}\)

The revenue \(R\) and cost \(C\) for a product are given by \(R=x(75-0.0005 x)\) and \(C=30 x+250,000\), where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold (see figure). (a) How many units must be sold to obtain a profit of at least \(\$ 750,000 ?\) (b) The demand equation for the product is \(p=75-0.0005 x\) where \(p\) is the price per unit. What prices will produce a profit of at least \(\$ 750,000 ?\) (c) As the number of units increases, the revenue eventually decreases. After this point, at what number of units is the revenue approximately equal to the cost? How should this affect the company's decision about the level of production?

Solve the inequality. Then graph the solution set on the real number line. \(|x+14|+3>17\)

A sales representative describes a "guaranteed investment fund" that is offered to new investors. You are told that if you deposit \(\$ 15,000\) in the fund you will be guaranteed to receive a total of at least \(\$ 40,000\) after 20 years. (a) If after 20 years you received the minimum guarantee, what annual interest rate did you receive? (b) If after 20 years you received \(\$ 48,000\), what annual interest rate did you receive? (Assume that the interest in the fund is compounded quarterly.)

Geometry A billboard is 10 feet longer than it is high (see figure). The billboard has 336 square feet of advertising space. What are the dimensions of the billboard?

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(\frac{x}{2.625}+\frac{x}{4.875}=1\)

\(P\) dollars, invested at interest rate \(r\) compounded annually, increases to an amount \(A=P(1+r)^{3}\) in 3 years. For an investment of \(\$ 1000\) to increase to an amount greater than \(\$ 1500\) in 3 years, the interest rate must be greater than what percent?

Solve the inequality. Then graph the solution set on the real number line. \(2|x+10| \geq 9\)

Geometry A triangular sign has a height that is equal to its base. The area of the sign is 4 square feet. Find the base and height of the sign.

Projected Expenses From January through May, a company's expenses totaled \(\$ 325,450\). If the monthly expenses continue at this rate, what will be the total expenses for the year?

\(P\) dollars, invested at interest rate \(r\) compounded annually, increases to an amount \(A=P(1+r)^{2}\) in 2 years. For an investment of \(\$ 2000\) to increase to an amount greater than \(\$ 2350\) in 2 years, the interest rate must be greater than what percent?

Solve the inequality. Then graph the solution set on the real number line. \(3|4-5 x| \leq 9\)

You take out a cash advance of \(\$ 1000\) on a credit card. After 2 months, you owe \(\$ 1041.93\). The interest is compounded monthly. What is the annual interest rate for this cash advance?

The per capita income \(P\) (in dollars) in the United States from 1995 to 2005 can be approximated by the model \(P=7.14 t^{2}+887.5 t+15,544\), \(5 \leq t \leq 15\), where \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) The figure shows the actual per capita income and the per capita income represented by the model. (a) Use the model to estimate the year in which the per capita income was about \(\$ 26,500\). (b) Use the model to predict the year in which the per capita income is about \(\$ 34,000\).

Projected Revenue From January through August, a company's revenues totaled \(\$ 549,680 .\) If the monthly revenue continues at this rate, what will be the total revenue for the year?

In Exercises 61-66, your answers are rounded to three decimal places. What effect does rounding have as you check a solution?

Solve the inequality. Then graph the solution set on the real number line. \(|x-5|<0\)

An airline offers daily flights between Chicago and Denver. The total monthly cost \(C\) (in millions of dollars) of these flights is modeled by \(C=\sqrt{0.25 x+1}\) where \(x\) is the number of passengers flying that month in thousands (see figure). The total cost of the flights for a month is \(3.5\) million dollars. Use the model to determine how many passengers flew that month.

Doctors treated a patient at an emergency room from 2:00 P.M. to 7:00 P.M. The patient's blood oxygen level \(L\) (in percent) during this time period can be modeled by $$ L=-0.270 t^{2}+3.59 t+83.1, \quad 2 \leq t \leq 7 $$ where \(t\) represents the time of day, with \(t=2\) corresponding to 2:00 P.M. Use the model to estimate the time (rounded to the nearest hour) when the patient's blood oxygen level was \(93 \%\).

Geometry A rectangular garden that is 30 feet long and 20 feet wide is surrounded on all four sides by a rock path that is \(x\) feet wide. The total area of the garden and the rock path is 1200 square feet. What is the width of the path?

Investment Mix You invest \(\$ 15,000\) in two funds paying \(6.5 \%\) and \(7.5 \%\) simple interest. The total annual interest is \(\$ 1020 .\) How much do you invest in each fund?

In Exercises 69-72, evaluate the expression in two ways. (a) Calculate entirely on your calculator using appropriate parentheses, and then round the answer to two decimal places. (b) Round both the numerator and the denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the second method introduce an additional roundoff error? $$ \frac{1+0.73205}{1-0.73205} $$

The average yearly cost \(C\) of higher education at public institutions in the United States for the academic years \(1995 / 1996\) to \(2004 / 2005\) can be modeled by \(C=30.57 t^{2}-259.6 t+6828, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to the \(1995 / 1996\) school year (see figure). Use the model to predict the academic year in which the average yearly cost of higher education at public institutions exceeds \(\$ 12,000\).

Solve the inequality. Then graph the solution set on the real number line. \(|x-5| \geq 0\)

The life expectancy of a person who is 16 to 25 years old can be modeled by \(y=\sqrt{1.244 x^{2}-161.16 x+6138.6}, \quad 16 \leq x \leq 25\) where \(y\) represents the number of additional years the person is expected to live and \(x\) represents the person's current age. (a) Determine the life expectancies of persons who are 18 , 20 , and 22 years old. (b) A person's life expectancy is 62 years. Use the model to determine the age of the person.

Doctors treated a patient at an emergency room from 2:00 P.M. to 7:00 P.M. The patient's blood oxygen level \(L\) (in percent) during this time period can be modeled by $$ L=-0.270 t^{2}+3.59 t+83.1, \quad 2 \leq t \leq 7 $$ where \(t\) represents the time of day, with \(t=2\) corresponding to 2:00 P.M. Use the model to estimate the time (rounded to the nearest hour) when the patient's blood oxygen level was \(93 \%\). 70\. Prescription Drugs The total amounts \(A\) (in billions of dollars) projected by the industry to be spent on prescription drugs in the United States from 2002 to 2012 can be approximated by the model. $$ A=0.89 t^{2}+15.9 t+126, \quad 2 \leq t \leq 12 $$ where \(t\) represents the year, with \(t=2\) corresponding to 2002\. Use the model to predict the year in which the total amount spent on prescription drugs will be about \(\$ 374,000,000,000\). (Source: U.S. Center for Medicine and Medicaid Services)

Geometry A rectangular pool is 30 feet wide and 40 feet long. It is surrounded on all four sides by a wooden deck that is \(x\) feet wide. The total area enclosed within the perimeter of the deck is 3000 square feet. What is the width of the deck?

Investment Mix You invest \(\$ 30,000\) in two funds paying \(3 \%\) and \(4 \frac{1}{2} \%\) simple interest. The total annual interest is \(\$ 1230\). How much do you invest in each fund?

Evaluate the expression in two ways. (a) Calculate entirely on your calculator using appropriate parentheses, and then round the answer to two decimal places. (b) Round both the numerator and the denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the second method introduce an additional roundoff error? $$ \frac{1+0.86603}{1-0.86603} $$

The average yearly cost \(C\) of higher education at private institutions in the United States for the academic years \(1995 / 1996\) to \(2004 / 2005\) can be modeled by \(C=42.93 t^{2}+68.0 t+15,309, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to the academic year \(1995 / 1996\) (see figure). Use the model to predict the academic year in which the average yearly cost of higher education at private institutions exceeds \(\$ 32,000\).

The life expectancy of a person who is 48 to 65 years old can be modeled by \(y=\sqrt{0.874 x^{2}-140.07 x+5752.5}, \quad 48 \leq x \leq 65\) where \(y\) represents the number of additional years the person is expected to live and \(x\) represents the person's current age. A person's life expectancy is 20 years. How old is the person?

Two planes leave simultaneously from the same airport, one flying due east and the other due south (see figure). The eastbound plane is flying 50 miles per hour faster than the southbound plane. After 3 hours the planes are 2440 miles apart. Find the speed of each plane.

In Exercises 71-76, 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 Falling Object A rock is dropped from the top of a 200 -foot cliff that overlooks the ocean. How long will it take for the rock to hit the water?

Stock Mix You invest \(\$ 5000\) in two stocks. In one year, the value of stock A increases by \(9.8 \%\) and the value of stock B increases by \(6.2 \%\). The total value of the stocks is now \(\$ 5389.20 .\) How much did you originally invest in each stock?

Evaluate the expression in two ways. (a) Calculate entirely on your calculator using appropriate parentheses, and then round the answer to two decimal places. (b) Round both the numerator and the denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the second method introduce an additional roundoff error? $$ \frac{333+\frac{1.98}{0.74}}{4+\frac{6.25}{3.15}} $$

Two planes leave simultaneously from the same airport, one flying due east and the other due south. The eastbound plane is flying 100 miles per hour faster than the southbound plane. After 2 hours the planes are 1500 miles apart. Find the speed of each plane.

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 Royal Gorge Bridge The Royal Gorge Bridge near Canon City, Colorado is the highest suspension bridge in the world. The bridge is 1053 feet above the Arkansas river. A rock is dropped from the bridge. How long does it take the rock to hit the water?

Stock Mix You invest \(\$ 4000\) in two stocks. In one year, the value of stock A increases by \(5.4 \%\) and the value of stock B increases by \(12.8 \%\). The total value of the stocks is now \(\$ 4401\). How much did you originally invest in each stock?

Evaluate the expression in two ways. (a) Calculate entirely on your calculator using appropriate parentheses, and then round the answer to two decimal places. (b) Round both the numerator and the denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the second method introduce an additional roundoff error? $$ \frac{1.73205-1.19195}{3-(1.73205)(1.19195)} $$

When two resistors of resistances \(R_{1}\) and \(R_{2}\) are connected in parallel (see figure), the total resistance \(R\) satisfies the equation \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) Find \(R_{1}\) for a parallel circuit in which \(R_{2}=2\) ohms and \(R\) must be at least 1 ohm.

The demand equation for a product is modeled by \(p=40-\sqrt{0.01 x+1}\), where \(x\) is the number of units demanded per day and \(p\) is the price per unit. Find the demand when the price is set at \(\$ 13.95 .\) Explain why this model is only valid for \(0 \leq x \leq 159,900\)

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 Olympic Diver The high-dive platform in the Olympics is 10 meters above the water. A diver wants to perform an armstand dive, which means she will drop to the water from a handstand position. How long will the diver be in the air? (Hint: 1 meter \(\approx 3.2808\) feet)

Comparing Investment Returns You invest \(\$ 12,000\) in a fund paying \(9 \frac{1}{2} \%\) simple interest and \(\$ 8000\) in a fund for which the interest rate varies. At the end of the year the total interest for both funds is \(\$ 2054.40 .\) What simple interest rate yields the same interest amount as the variable rate fund?