Chapter 1

MARKUP ON A CAR The markup on a used car was at least \(30 \%\) of its current wholesale price. If the car was sold for \(\$ 6500\), what was the maximum wholesale price?

Solve the equation. $$ \sqrt{6 x^{2}-5 x}-2=0 $$

Rationalize the denominator of the expression. $$ \frac{5 x^{2}}{\sqrt{3 x}} $$

The total revenue realized by the Apollo Company from the sale of \(x\) PDAs is given by \(R(x)=-0.1 x^{2}+500 x\) dollars. Factor the expression on the right- hand side of this equation.

A manufacturer of tennis rackets finds that the total cost of manufacturing \(x\) rackets/day is given by $$ 0.0001 x^{2}+4 x+400 $$ dollars. Each racket can be sold at a price of \(p\) dollars, where $$ p=-0.0004 x+10 $$ Find an expression giving the daily profit for the manufacturer, assuming that all the rackets manufactured can be sold.

A manufacturer of a certain commodity has estimated that her profit (in thousands of dollars) is given by the expression $$ -6 x^{2}+30 x-10 $$ where \(x\) (in thousands) is the number of units produced. What production range will enable the manufacturer to realize a profit of at least \(\$ 14,000\) on the commodity?

Solve the equation. $$ \sqrt{2 r+3}=r $$

Rationalize the denominator of the expression. $$ \frac{1}{\sqrt[3]{x}} $$

The 1980 s saw a trend toward old-fashioned punitive deterrence as opposed to the more liberal penal policies and community-based corrections popular in the \(1960 \mathrm{~s}\) and early \(1970 \mathrm{~s}\). As a result, prisons became more crowded, and the gap between the number of people in prison and prison capacity widened. Based on figures from the U.S. Department of Justice, the number of prisoners (in thousands) in federal and state prisons is approximately $$ 3.5 t^{2}+26.7 t+436.2 \quad(0 \leq t \leq 10) $$ and the number of inmates (in thousands) for which prisons were designed is given by $$ 24.3 t+365 \quad(0 \leq t \leq 10) $$ where \(t\) is measured in years and \(t=0\) corresponds to 1984 . Find an expression giving the gap between the number of prisoners and the number for which the prisons were designed at any time \(t .\)

CONCENTRATION OF A DRUG IN THE BLOODSTREAM The concentration (in milligrams/cubic centimeter) of a certain drug in a patient's bloodstream \(t\) hr after injection is given by $$ \frac{0.2 t}{t^{2}+1} $$ Find the interval of time when the concentration of the drug is greater than or equal to \(0.08 \mathrm{mg} / \mathrm{cc}\).

Solve the equation. $$ \sqrt{3-4 x}+2 x=0 $$

Rationalize the denominator of the expression. $$ \sqrt{\frac{2 x}{y}} $$

Health-care spending per person (in dollars) by the private sector includes payments by individuals, corporations, and their insurance companies and is approximated by $$ 2.5 t^{2}+18.5 t+509 \quad(0 \leq t \leq 6) $$ where \(t\) is measured in years and \(t=0\) corresponds to the beginning of 1994 . The corresponding government spending (in dollars), including expenditures for Medicaid and other federal, state, and local government public health care, is $$ -1.1 t^{2}+29.1 t+429 \quad(0 \leq t \leq 6) $$ where \(t\) has the same meaning as before. Find an expression for the difference between private and government expenditures per person at any time \(t .\) What was the difference between private and government expenditures per person at the beginning of 1998 ? At the beginning of 2000 ?

A city's main well was recently found to be contaminated with trichloroethylene (a cancer-causing chemical) as a result of an abandoned chemical dump that leached chemicals into the water. A proposal submitted to the city council indicated that the cost, in millions of dollars, of removing \(x \%\) of the toxic pollutants is $$ \frac{0.5 x}{100-x} $$ If the city could raise between \(\$ 25\) and \(\$ 30\) million inclusive for the purpose of removing the toxic pollutants, what is the range of pollutants that could be expected to be removed?

Rationalize the denominator of the expression. $$ \frac{2}{1+\sqrt{3}} $$

The average speed of a vehicle in miles per hour on a stretch of route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the expression $$ 20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4) $$ where \(t\) is measured in hours, with \(t=0\) corresponding to 6 a.m. Over what interval of time is the average speed of a vehicle less than or equal to \(35 \mathrm{mph}\) ?

Solve the equation. $$ \sqrt{x+1}-\sqrt{2 x-5}+1=0 $$

Rationalize the denominator of the expression. $$ \frac{3}{1-\sqrt{2}} $$

EFFECT OF BACTERICIDE The number of bacteria in a certain culture \(t\) min after an experimental bactericide is introduced is given by $$ \frac{10,000}{t^{2}+1}+2000 $$ Find the time when the number of bacteria will have dropped below 4000 .

In Exercises \(63-70\), use the discriminant to determine the number of real solutions of the equation. $$ x^{2}-6 x+5=0 $$

Rationalize the denominator of the expression. $$ \frac{1+\sqrt{2}}{1-\sqrt{2}} $$

Nitrogen dioxide is a brown gas that impairs breathing. The amount of nitrogen dioxide present in the atmosphere on a certain May day in the city of Long Beach measured in PSI (pollutant standard index) at time \(t\). where \(t\) is measured in hours, and \(t=0\) corresponds to 7 a.m., is approximated by $$ \frac{136}{1+0.25(t-4.5)^{2}}+28 \quad(0 \leq t \leq 11) $$ Find the time of the day when the amount of nitrogen diox-

Solve the equation. $$ 2 m^{2}+5 m+3=0 $$

Rationalize the denominator of the expression. $$ \frac{9+\sqrt{2}}{3-\sqrt{2}} $$

A ball is thrown straight up so that its height after \(t\) sec is $$ 128 t-16 t^{2}+4 $$ ft. Determine the length of time the ball stays above a height of \(196 \mathrm{ft}\).

Use the discriminant to determine the number of real solutions of the equation. $$ 3 y^{2}-4 y+5=0 $$

Rationalize the denominator of the expression. $$ \frac{q}{\sqrt{q}-1} $$

The distribution of income in a certain city can be described by the mathematical model \(y=\left(2.8 \cdot 10^{11}\right)(x)^{-1.5}\), where \(y\) is the number of families with an income of \(x\) or more dollars. a. How many families in this city have an income of \(\$ 20,000\) or more? b. How many families have an income of \(\$ 40,000\) or more? c. How many families have an income of \(\$ 100,000\) or more?

Use the discriminant to determine the number of real solutions of the equation. $$ 2 p^{2}+5 p+6=0 $$

Rationalize the denominator of the expression. $$ \frac{x y}{\sqrt{x}+\sqrt{y}} $$

Manufacturing Company manufactures steel rods. Suppose the rods ordered by a customer are manufactured to a specification of \(0.5\) in. and are acceptable only if they are within the tolerance limits of \(0.49\) in. and \(0.51\) in. Letting \(x\) denote the diameter of a rod, write an inequality using absolute value to express a criterion involving \(x\) that must be satisfied in order for a rod to be acceptable.

Use the discriminant to determine the number of real solutions of the equation. $$ 4 x^{2}+12 x+9=0 $$

Rationalize the denominator of the expression. $$ \frac{y}{\sqrt[3]{x^{2} z}} $$

The diameter \(x\) (in inches) of a batch of ball bearings manufactured by PAR Manufacturing satisfies the inequality $$ |x-0.1| \leq 0.01 $$ What is the smallest diameter a ball bearing in the batch can have? The largest diameter?

Use the discriminant to determine the number of real solutions of the equation. $$ 25 x^{2}-80 x+64=0 $$

Rationalize the denominator of the expression. $$ \frac{2 x}{\sqrt[3]{x y^{2}}} $$

Use the discriminant to determine the number of real solutions of the equation. $$ \frac{6}{k^{2}}+\frac{1}{k}-2=0 $$

Write the expression in simplest radical form. $$ \sqrt{\frac{16}{3}} $$

Use the discriminant to determine the number of real solutions of the equation. $$ (2 p+1)^{2}-3(2 p+1)+4=0 $$

Write the expression in simplest radical form. $$ -\sqrt{\frac{8}{3}} $$

A person standing on the balcony of a building throws a ball directly upward. The height of the ball as measured from the ground after \(t\) sec is given by \(h=-16 t^{2}+64 t+768\). When does the ball reach the ground?

Write the expression in simplest radical form. $$ \sqrt[3]{\frac{2}{3}} $$

A model rocket is launched vertically upward so that its height (measured in feet) \(t \mathrm{sec}\) after launch is given by $$ h(t)=-16 t^{2}+384 t+4 $$ a. Find the time(s) when the rocket is at a height of \(1284 \mathrm{ft}\). b. How long is the rocket in flight?

Write the expression in simplest radical form. $$ \sqrt[3]{\frac{81}{4}} $$

A cyclist riding along a straight path has a speed of \(u \mathrm{ft} / \mathrm{sec}\) as she passes a tree. Accelerating at \(a \mathrm{ft} / \mathrm{sec}^{2}\), she reaches a speed of \(v \mathrm{ft} / \mathrm{sec} t\) sec later, where \(v=u t+a t^{2}\). If the cyclist was traveling at \(10 \mathrm{ft} / \mathrm{sec}\) and she began accelerating at a rate of \(4 \mathrm{ft} / \mathrm{sec}^{2}\) as she passed the tree, how long did it take her to reach a speed of \(22 \mathrm{ft} / \mathrm{sec} ?\)

Write the expression in simplest radical form. $$ \sqrt{\frac{3}{2 x^{2}}} $$

Phillip, the proprietor of a vineyard estimates that the profit from producing and selling \((x+10,000)\) bottles of wine is \(P=-0.0002 x^{2}+3 x+\) 50,000 . Find the level(s) of production that will yield a profit of \(\$ 60,800\).

Write the expression in simplest radical form. $$ \sqrt{\frac{x^{3} y^{5}}{4}} $$

The quantity demanded \(x\) (measured in units of a thousand) of the Sentinel smoke alarm/week is related to its unit price \(p\) (in dollars) by the equation $$ p=\frac{30}{0.02 x^{2}+1} \quad(0 \leq x \leq 10) $$ If the unit price is set at \(\$ 10\), what is the quantity demanded?

Write the expression in simplest radical form. $$ \sqrt[3]{\frac{2 y^{2}}{3}} $$