Chapter 3

A rectangular pen for a pet is under construction using 100 feet of fence. (a) Find the dimensions that give an area of 576 square feet. (b) Find the dimensions that give maximum area.

A cylindrical aluminum can is being constructed to have a height \(h\) of 4 inches. If the can is to have a volume of 28 cubic inches, approximate its radius \(r .\) (Hint: \(V=\pi r^{2} h\).)

Braking distance for cars on level pavement can be approximated by \(D(x)=\frac{x^{2}}{30 k^{*}}\) The input \(x\) is the car's velocity in miles per hour and the output \(D(x)\) is the braking distance in feet. The positive constant \(k\) is a measure of the traction of the tires. Small values of \(k\) indicate a slippery road or worn tires. (Source: L. Haefner, Introduction to Transportation Systems.) (a) Let \(k=0.3 .\) Evaluate \(D(60)\) and interpret the result. (b) If \(k=0.25,\) find the velocity \(x\) that corresponds to a braking distance of 300 feet.

A window comprises a square with sides of length \(x\) and a semicircle with diameter \(x\) as shown in the figure. If the total area of the window is 463 square inches, estimate the value of \(x\) to the nearest hundredth of an inch. (PICTURE NOT COPY)

Aframe for a picture is 2 inches wide. The picture inside the frame is 4 inches longer than it is wide. See the figure. If the area of the picture is 320 square inches, find the outside dimensions of the picture frame. (PICTURE NOT COPY)

(Refer to Example 12.) A company charges \(\$ 20\) to make one monogrammed shirt, but reduces this cost by \(\$ 0.10\) per shirt for each additional shirt ordered up to 100 shirts. If the cost of an order is \(\$ 989,\) how many shirts were ordered?

One airline ticket costs \(\$ 250\). For each additional airline ticket sold to a group, the price of every ticket is reduced by \(\$ 2 .\) For example, 2 tickets \(\operatorname{cost} 2 \cdot 248=\$ 496\) and 3 tickets \(\operatorname{cost} 3 \cdot 246=\$ 738\) (a) Write a quadratic function that gives the total cost of buying \(x\) tickets. (b) What is the cost of 5 tickets? (c) How many tickets were sold if the cost is \(\$ 5200 ?\) (d) What number of tickets sold gives the greatest cost?

The table lists the velocity and distance raveled by a falling object for various elapsed times. $$ \begin{array}{|rcccccc|} \hline \text { Time (sec) } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Velocity (fl/sec) } & 0 & 32 & 64 & 96 & 128 & 160 \\ \hline \text { Distance (ft) } & 0 & 16 & 64 & 144 & 256 & 400 \\ \hline \end{array} $$ (a) Make a scatterplot of the ordered pairs determined by (time, velocity) and (time, distance) in the same vicwing rectangle \([-1,6,1]\) by \([-10,450,50]\) (b) Find a function \(v\) that models the velocity. (c) The distance is modeled by \(d(x)=a x^{2} .\) Find \(a\) (d) Find the time when the distance is 200 feet. Find the velocity at this time.

The road (or taxiway) used by aircraft to exit a runway should not have sharp curves. The safe radius for any curve depends on the speed of the airplane. The table at the top of the next column lists the minimum radius \(R\) of the exit curves, where the taxiing speed of the airplane is \(x\) miles per hour. $$ \begin{array}{rllllll} \hline x(\mathrm{miflar}) & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline R(\mathrm{ft}) & 50 & 200 & 450 & 800 & 1250 & 1800 \\ \hline \end{array} $$ (a) If the taxiing speed \(x\) of the plane doubles, what happens to the minimum radius \(R\) of the curve? (b) The FAA used \(R(x)=a x^{2}\) to compute the values in the table. Determine \(a\). (c) If \(R=500,\) find \(x\). Interpret your results.

Some types of worms have a remarkable capacity to live without moisture. The table shows the number of worms \(y\) surviving after \(x\) days in one study. $$ \begin{array}{cccccc} x \text { (days) } & 0 & 20 & 40 & 80 & 120 \\ y \text { (worms) } & 50 & 48 & 45 & 36 & 20 \end{array} $$ (a) Use regression to find a quadratic function \(f\) that models these data. (b) Graph \(f\) and the data in the same window. (c) Solve the quadratic equation \(f(x)=0\) graphically. Do both solutions have meaning? Explain.

The table lists numbers of Wal-Mart employees \(E\) in millions, \(x\) years after 1987 . $$ \begin{array}{cccccc} x & 0 & 5 & 10 & 15 & 20 \\ \hline B & 0.20 & 0.38 & 0.68 & 1.4 & 2.2 \end{array} $$ (a) Evaluate \(E(15)\) and interpret the result. (b) Find a quadratic function \(f\) that models these data. (c) Graph the data and function \(f\) in the same \(x y\) -plane. (d) Use \(f\) to estimate the year when the number of cmployees may reach 3 million.

The number \(N\) of women in millions who were gainfully employed in the work force in selected years is shown in the table. $$ \begin{array}{|rccccccc} \hline \text { Year } & 1900 & 1910 & 1920 & 1930 & 1940 & 1950 \\ \hline N & 5.3 & 7.4 & 8.6 & 10.8 & 12.8 & 18.4 \\ \hline \text { Year } & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\ \hline N & 23.2 & 31.5 & 45.5 & 56.6 & 65.6 & 74.8 \end{array} $$ (a) Use regression to find a quadratic function \(f\) that models the data. Support your result graphically. (b) Predict the number of women in the labor force in 2020

Discuss three symbolic methods for solving a quadratic equation. Make up a quadratic equation and use each method to find the solution set.

Explain how to solve a quadratic equation graphically.