Chapter 1

Gas Mileage The gas mileage \(g\) (measured in mi/gal) for a particular vehicle, driven at \(v\) mi/h, is given by the formula \(g=10+0.9 v-0.01 v^{2},\) as long as \(v\) is between 10 mi/h and 75 \(\mathrm{mi} / \mathrm{h}\) . For what range of speeds is the vehicle's mileage 30 \(\mathrm{mi} / \mathrm{gal}\) or better?

A cylindrical can has a volume of 40\(\pi \mathrm{cm}^{3}\) and is 10 \(\mathrm{cm}\) tall. What is its diameter? [Hint: Use the volume formula listed on the inside front cover of this book.

\(79-92\) Solve the equation for the indicated variable. $$ a-2[b-3(c-x)]=6 ; \text { for } x $$

Stopping Distance For a certain model of car the distance \(d\) required to stop the vehicle if it is traveling at \(v\) mi/h is given by the formula $$ d=v+\frac{v^{2}}{20} $$ where \(d\) is measured in feet. Kerry wants her stopping distance not to exceed 240 \(\mathrm{ft}\) . At what range of speeds can she travel?

A parcel of land is 6 \(\mathrm{ft}\) longer than it is wide. Each diagonal from one corner to the opposite corner is 174 \(\mathrm{ft}\) long. What are the dimensions of the parcel?

\(79-92\) Solve the equation for the indicated variable. $$ a^{2} x+(a-1)=(a+1) x ; \text { for } x $$

Manufacturer's Profit If a manufacturer sells \(x\) units of a certain product, his revenue \(R\) and cost \(C\) (in dollars) are given by: $$ \begin{array}{l}{R=20 x} \\ {C=2000+8 x+0.0025 x^{2}}\end{array} $$ Use the fact that profit \(=\) revenue \(-\) cost to determine how many units he should sell to enjoy a profit of at least \(\$ 2400 .\)

A flagpole is secured on opposite sides by two guy wires, each of which is 5 \(\mathrm{ft}\) longer than the pole. The distance between the points where the wires are fixed to the ground is equal to the length of one guy wire. How tall is the flagpole (to the nearest inch)?

\(79-92\) Solve the equation for the indicated variable. $$ \frac{a+1}{b}=\frac{a-1}{b}+\frac{b+1}{a} ; \text { for } a $$

Theater Tour Cost A riverboat theater offers bus tours to groups on the following basis. Hiring the bus costs the group \(\$ 360\) , to be shared equally by the group members. Theater tickets, normally \(\$ 30\) each, are discounted by 25\(€\) times the number of people in the group. How many members must be in the group so that the cost of the theater tour (bus fare plus theater ticket) is less than \(\$ 39\) per person?

Suppose an object is dropped from a height \(h_{0}\) above the ground. Then its height after \(t\) seconds is given by \(h=-16 t^{2}+h_{0},\) where \(h\) is measured in feet. Use this information to solve the problem. If a ball is dropped from 288 ft above the ground, how long does it take to reach ground level?

\(79-92\) Solve the equation for the indicated variable. $$ V=\frac{1}{3} \pi r^{2} h ; \quad \text { for } r $$

Fencing a Garden \(\quad\) A determined gardener has 120 \(\mathrm{ft}\) of deer- resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area enclosed to be at least 800 \(\mathrm{ft}^{2} .\) What range of values is possible for the length of her garden?

Suppose an object is dropped from a height \(h_{0}\) above the ground. Then its height after \(t\) seconds is given by \(h=-16 t^{2}+h_{0},\) where \(h\) is measured in feet. Use this information to solve the problem. A ball is dropped from the top of a building 96 \(\mathrm{ft}\) tall. (a) How long will it take to fall half the distance to ground level? (b) How long will it take to fall to ground level?

\(79-92\) Solve the equation for the indicated variable. $$ F=G \frac{m M}{r^{2}} ; \quad \text { for } r $$

Do Powers Preserve Order? If \(a

Use the formula \(h=-16 t^{2}+v_{0} t\) discussed in Example 7. A ball is thrown straight upward at an initial speed of \(v_{0}=40 \mathrm{ft} / \mathrm{s}\). (a) When does the ball reach a height of 24 \(\mathrm{ft}\) ? (b) When does it reach a height of 48 \(\mathrm{ft}\) ? (c) What is the greatest height reached by the ball? (d) When does the ball reach the highest point of its path? (e) When does the ball hit the ground?

\(79-92\) Solve the equation for the indicated variable. $$ a^{2}+b^{2}=c^{2} ; \quad \text { for } b $$

What's Wrong Here? It is tempting to try to solve an inequality like an equation. For instance, we might try to solve \(1<3 / x\) by multiplying both sides by \(x,\) to get \(x<3\) so the solution would be \((-\infty, 3) .\) But that's wrong; for example, \(x=-1\) lies in this interval but does not satisfy the original inequality. Explain why this method doesn't work (think about the sign of \(x\) ). Then solve the inequality correctly.

Use the formula \(h=-16 t^{2}+v_{0} t\) discussed in Example 7. How fast would a ball have to be thrown upward to reach a maximum height of 100 \(\mathrm{ft}\) ? [Hint: Use the discriminant of the equation \(16 t^{2}-v_{0} t+h=0.\)]

\(79-92\) Solve the equation for the indicated variable. $$ A=P\left(1+\frac{i}{100}\right)^{2} ; \quad \text { for } i $$

Ordering the Points in the Cartesian Plane Given any two different real numbers \(x\) and \(y,\) we know that either \(x

The fish population in a certain lake rises and falls according to the formula $$F=1000\left(30+17 t-t^{2}\right)$$ Here \(F\) is the number of fish at time \(t\) , where \(t\) is measured in years since January \(1,2002,\) when the fish population was first estimated. (a) On what date will the fish population again be the same as on January \(1,2002 ?\) (b) By what date will all the fish in the lake have died?

\(79-92\) Solve the equation for the indicated variable. $$ V=\frac{4}{3} \pi r^{3} ; \quad \text { for } r $$

Comparing Areas A wire 360 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

A salesman drives from Ajax to Barrington, a distance of 120 \(\mathrm{mi}\) , at a steady speed. He then increases his speed by 10 \(\mathrm{mi} / \mathrm{h}\) to drive the 150 \(\mathrm{mi}\) from Barrington to Collins. If the second leg of his trip took 6 min more time than the first leg, how fast was he driving between Ajax and Barrington?

Shrinkage in Concrete Beams As concrete dries, it shrinks-the higher the water content, the greater the shrinkage. If a concrete beam has a water content of \(w \mathrm{kg} / \mathrm{m}^{3}\) , then it will shrink by a factor $$S=\frac{0.032 w-2.5}{10,000}$$ where \(S\) is the fraction of the original beam length that disappears due to shrinkage. (a) A beam 12.025 \(\mathrm{m}\) long is cast in concrete that contains 250 \(\mathrm{kg} / \mathrm{m}^{3}\) water. What is the shrinkage factor \(S ?\) How long will the beam be when it has dried? (b) \(\mathrm{A}\) beam is 10.014 \(\mathrm{m}\) long when wet. We want it to shrink to \(10.009 \mathrm{m},\) so the shrinkage factor should be \(S=0.00050 .\) What water content will provide this amount of shrinkage?

Kiran drove from Tortula to Cactus, a distance of 250 mi. She increased her speed by 10 milh for the \(360-\) mi trip from Cactus to Dry Junction. If the total trip took 11 \(\mathrm{h}\) , what was her speed from Tortula to Cactus?

Manufacturing Cost A toy maker finds that it costs \(C=450+3.75 x\) dollars to manufacture \(x\) toy trucks. If the budget allows \(\$ 3600\) in costs, how many trucks can be made?

It took a crew 2 h 40 \(\mathrm{min}\) to row 6 \(\mathrm{km}\) upstream and back again. If the rate of flow of the stream was 3 \(\mathrm{km} / \mathrm{h}\) , what was the rowing speed of the crew in still water?

Power Produced by a Windmill When the wind blows with speed \(v \mathrm{km} / \mathrm{h}\) , a windmill with blade length 150 \(\mathrm{cm}\) generates \(P\) watts \((\mathrm{W})\) of power according to the formula \(P=15.6 v^{3} .\) (a) How fast would the wind have to blow to generate \(10,000 \mathrm{W}\) of power? (b) How fast would the wind have to blow to generate \(50,000 \mathrm{W}\) of power?

Two fishing boats depart a harbor at the same time, one traveling east, the other south. The east-bound boat travels at a speed 3 \(\mathrm{mi} / \mathrm{h}\) faster than the south-bound boat. After two hours the boats are 30 \(\mathrm{mi}\) apart. Find the speed of the southbound boat.

Food Consumption The average daily food consumption \(F\) of a herbivorous mammal with body weight \(x,\) where both \(F\) and \(x\) are measured in pounds, is given approximately by the equation \(F=0.3 x^{3 / 4} .\) Find the weight \(x\) of an elephant who consumes 300 lb of food per day.

A factory is to be built on a lot measuring 180 \(\mathrm{ft}\) by 240 \(\mathrm{ft}\) . A local building code specifies that a lawn of uniform width and equal in area to the factory must surround the factory. What must the width of this lawn be, and what are the dimensions of the factory?

A 19\(\frac{1}{2}\) -foot ladder leans against a building. The base of the ladder is 7\(\frac{1}{2}\) ft from the building. How high up the building does the ladder reach?

Proof That \(0=1 ?\) The following steps appear to give equivalent equations, which seem to prove that \(1=0 .\) Find the error. $$ \begin{aligned} x &=1 \\ x^{2} &=x \\ x^{2}-x &=0 \\ x(x-1) &=0 \\ x &=0 \\\ 1 &=0 \end{aligned} $$

Henry and Irene working together can wash all the windows of their house in 1 \(\mathrm{h} 48 \mathrm{min}\) . Work- ing alone, it takes Henry 1\(\frac{1}{2} \mathrm{h}\) more than Irene to do the job. How long does it take each person working alone to wash all the windows?

Jack, Kay, and Lynn deliver advertising flyers in a small town. If each person works alone, it takes Jack 4 \(\mathrm{h}\) to deliver all the flyers, and it takes Lynn 1 \(\mathrm{h}\) longer than it takes Kay. Working together, they can deliver all the flyers in 40\(\%\) of the time it takes Kay working alone. How long does it take Kay to deliver all the flyers alone?

If an imaginary line segment is drawn between the centers of the earth and the moon, then the net gravitational force \(F\) acting on an object situated on this line segment is $$F=\frac{-K}{x^{2}}+\frac{0.012 K}{(239-x)^{2}}$$ where \(K>0\) is a constant and \(x\) is the distance of the object from the center of the earth, measured in thousands of miles. How far from the center of the earth is the "dead spot' where no net gravitational force acts upon the object? (Express your answer to the nearest thousand miles.)

The quadratic formula gives us the roots of a quadratic equation from its coefficients. We can also obtain the coefficients from the roots. For example, find the roots of the equation \(x^{2}-9 x+20=0\) and show that the product of the roots is the constant term 20 and the sum of the roots is 9 , the negative of the coefficient of \(x .\) Show that the same relationship between roots and coefficients holds for the following equations: $$\begin{array}{l}{x^{2}-2 x-8=0} \\ {x^{2}+4 x+2=0}\end{array}$$ Use the quadratic formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has roots \(r_{1}\) and \(r_{2},\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)

The ancient Babylonians knew how to solve quadratic equations. Here is a problem from a cuneiform tablet found in a Babylonian school dating back to about 2000 \(\mathrm{B.C.}\) I have a reed, I know not its length. I broke from it one cubit, and it fit 60 times along the length of my field. I restored to the reed what I had broken off, and it fit 30 times along the width of my field. The area of my field is 375 square nindas. What was the original length of the reed? Solve this problem. Use the fact that 1 ninda \(=12\) cubits.