Chapter 1

Falling-Body Problems 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?

What's Wrong Here? It is tempting to try to solve an in- equality as if it were 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 in- equality. Explain why this method doesn't work (think about the sign of \(x\) ). Then solve the inequality correctly.

Falling-Body Problems 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?

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 owing 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) 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?

Falling-Body Problems 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 \(\mathrm{ft}\) ? \([\text { Hint: Use the discriminant of }\) the equation \(16 t^{2}-v_{0} t+h=0.1\)

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?

Fish Population 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 it was on January \(1,2002 ?\) (b) By what date will all the fish in the lake have died?

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 \(P=15.6 \mathrm{v}^{3}\). (a) How fast would the wind have to blow to generate 10,000 W of power? (b) How fast would the wind have to blow to generate 50,000 W of power?

Comparing Areas A wire 360 in. long is cut into two pieces. One piece is formed into a square, and the other is formed 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)?

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 that consumes 300 lb of food per day.

Width of a Lawn 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?

The equation $$3 x+k-5=k x-k+1$$ is really a family of equations, because for each value of \(k\), we get a different equation with the unknown \(x\). The letter \(k\) is called a parameter for this family. What value should we pick for \(k\) to make the given value of \(x\) a solution of the resulting equation? (a) \(x=0 \quad\) (b) \(x=1 \quad\) (c) \(x=2\)

Reach of a Ladder \(A 19 \frac{1}{2}\) -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?

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

Sharing a Job Henry and Irene working together can wash all the windows of their house in 1 168 min. Working alone, it takes Henry 1\(\frac{1}{2}\) h more than Irene to do the job. How long does it take each person working alone to wash all the windows?

Sharing a Job 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?

Gravity 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.)

Relationship Between Roots and Coefficients 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 \(b=-\left(r_{1}+r_{2}\right)\)