Chapter 3

Optimal Area An indoor physical-fitness room consists of a rectangular region with a semicircle on each end (see figure). The perimeter of the room is to be a 200-meter running track. What measurements will produce a maximum area of the rectangle?

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$h(t)=\frac{3 t^{2}}{t^{2}-4}$$

Write the quotient in standard form. $$\frac{8+15 i}{3 i}$$

Dimensions of a Box An open box is to be made from a rectangular piece of material, 16 inches by 12 inches, by cutting equal squares from the corners and tuming up the sides (see figure). (a) Write the volume \(V\) of the box as a function of \(x\). Determine the domain of the function. (b) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume. (c) Find values of \(x\) such that \(V=120\). Which of these values is a physical impossibility in the construction of the box? Explain. (d) What value of \(x\) should you use to make the tallest possible box with a volume of 120 cubic inches?

Use synthetic division to find each function value. \(f(x)=0.4 x^{4}-1.6 x^{3}+0.7 x^{2}-2\) (a) \(f(1)\) (b) \(f(-2)\) (c) \(f(5)\) (d) \(f(-10)\)

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=4 x^{2}-x^{3}$$

Find the number of units that produces a maximum revenue. The revenue \(R\) is measured in dollars and \(x\) is the number of units produced. $$R=1000 x-0.02 x^{2}$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$g(s)=\frac{s}{s^{2}+1}$$

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=x^{4}-2 x^{3}+37 x^{2}-72 x+36, \quad 6 i$$

Write the quotient in standard form. $$\frac{1}{(2 i)^{3}}$$

Dimensions of a Terrarium A rectangular terrarium with a square cross section has a combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the terrarium, given that the volume is 11,664 cubic inches.

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((x+2),(x-4)\) Function $$f(x)=x^{3}-12 x-16$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=1-x^{3}$$

Find the number of units that produces a maximum revenue. The revenue \(R\) is measured in dollars and \(x\) is the number of units produced. $$R=80 x-0.0001 x^{2}$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$g(x)=\frac{x}{x^{2}+3}$$

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=x^{3}-7 x^{2}-x+87, \quad 5+2 i$$

Write the quotient in standard form. $$\frac{1}{(3 i)^{3}}$$

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((x+4),(x-6)\) Function $$f(x)=x^{3}-28 x-48$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=x^{3}-9 x$$

Optimal Cost A manufacturer of lighting fixtures has daily production costs \(C\) (in dollars per unit) of \(C(x)=800-10 x+0.25 x^{2}\) where \(x\) is the number of units produced. How many fixtures should be produced each day to yield a minimum cost per unit?

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{x}{x^{2}-3 x-4}$$

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=4 x^{3}+23 x^{2}+34 x-10,-3+i$$

Write the quotient in standard form. $$\frac{4}{(1-2 i)^{3}}$$

Geometry A bulk food storage bin with dimensions 2 feet by 3 feet by 4 feet needs to be increased in size to hold five times as much food as the current bin. (Assume each dimension is increased by the same amount.) (a) Write a function that represents the volume \(V\) of the new bin. (b) Find the dimensions of the new bin.

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((3 x+1),(x-2)\) Function $$f(x)=3 x^{3}+10 x^{2}-27 x-10$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=\frac{1}{4} x^{4}-2 x^{2}$$

Optimal Profit The profit \(P\) (in dollars) for a manufacturer of sound systems is given by \(P(x)=-0.0003 x^{2}+150 x-375,000\) where \(x\) is the number of units produced. What production level will yield a maximum profit?

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{-x}{x^{2}+x-6}$$

Write the quotient in standard form. $$\frac{3}{(5-2 i)^{2}}$$

Geometry A rancher wants to enlarge an existing rectangular corral such that the total area of the new corral is \(1.5\) times that of the original corral. The current corral's dimensions are 250 feet by 160 feet. The rancher wants to increase each dimension by the same amount. (a) Write a function that represents the area \(A\) of the new corral. (b) Find the dimensions of the new corral. (c) A rancher wants to add a length to the sides of the corral that are 160 feet, and twice the length to the sides that are 250 feet, such that the total area of the new corral is \(1.5\) times that of the original corral. Repeat parts (a) and (b). Explain your results.

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((5 x-1),(x-4)\) Function $$f(x)=5 x^{3}-11 x^{2}-38 x+8$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$g(t)=-\frac{1}{4}(t-2)^{2}(t+2)^{2}$$

Maximum Height of a Diver The path of a diver is given by \(y=-\frac{4}{9} x^{2}+\frac{24}{9} x+10\) where \(y\) is the height (in feet) and \(x\) is the horizontal distance from the end of the diving board (in feet) (see figure). Use a graphing utility and the trace or maximum feature to find the maximum height of the diver.

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{3 x}{x^{2}-x-2}$$

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22,-3+\sqrt{2} i$$

Write the quotient in standard form. $$\frac{(21-7 i)(4+3 i)}{2-5 i}$$

Medicine The concentration \(C\) of a chemical in the bloodstream \(t\) hours after injection into muscle tissue is given by \(C=\frac{3 t^{2}+t}{t^{3}+50}, \quad t \geq 0\) The concentration is greatest when \(3 t^{4}+2 t^{3}-300 t-50=0\) Approximate this time to the nearest hundredth of an hour.

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((x-\sqrt{3}),(x+2)\) Function $$f(x)=x^{3}+2 x^{2}-3 x-6$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=x(x-2)^{2}(x+1)$$

Maximum Height The winning men's shot put in the 2004 Summer Olympics was recorded by Yuriy Belonog of Ukraine. The path of his winning toss is approximately given by \(y=-0.011 x^{2}+0.65 x+8.3\) where \(y\) is the height of the shot (in feet) and \(x\) is the horizontal distance (in feet). Use a graphing utility and the trace or maximum feature to find the length of the winning toss and the maximum height of the shot.

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{2 x}{x^{2}+x-2}$$

Write the quotient in standard form. $$\frac{(3-i)(2+5 i)}{4+3 i}$$

(a) verify the given factors of \(f(x)\), (b) find the remaining factor of \(f(x)\), (c) use your results to write the complete factorization of \(f(x)\), (d) list all real zeros of \(f\), and (e) confirm your results by using a graphing utility to graph the function. Factors \((x-\sqrt{2}),(x+2)\) Function $$f(x)=x^{3}+2 x^{2}-2 x-4$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=1-x^{6}$$

Cable TV Subscribers The table shows the average numbers \(S\) (in millions) of basic cable subscribers for the years 1995 to 2005. (Source: Kagan Research, LLC) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 \\ \hline \text { Subscribers, } S & 60.6 & 62.3 & 63.6 & 64.7 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Subscribers, } S & 65.5 & 66.3 & 66.7 & 66.5 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Subscribers, } S & 66.1 & 65.7 & 65.3 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=5\) corresponding to \(1995 .\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model from part (b) in the same viewing window as the scatter plot. (d) Use the graph of the model from part (c) to estimate when the number of basic cable subscribers was the greatest. Does this result agree with the actual data?

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) Vertical asymptote: None Horizontal asymptote: \(y=2\)

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=8 x^{3}-14 x^{2}+18 x-9, \quad \frac{1}{2}(1-\sqrt{5} i)$$

Online Sales The revenues per share \(R\) (in dollars) for Amazon.com for the years 1996 to 2005 are shown in the table. (Source: Amazon.com) $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{l} \text { Revenue per } \\ \text { share, } R \end{array} \\ \hline 1996 & 0.07 \\ \hline 1997 & 0.51 \\ \hline 1998 & 1.92 \\ \hline 1999 & 4.75 \\ \hline 2000 & 7.73 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{l} \text { Revenue per } \\ \text { share, } R \end{array} \\ \hline 2001 & 8.37 \\ \hline 2002 & 10.14 \\ \hline 2003 & 13.05 \\ \hline 2004 & 17.16 \\ \hline 2005 & 20.41 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to \(1996 .\) (b) Use the regression feature of a graphing utility to find a linear model, a quadratic model, a cubic model, and a quartic model for the data. (c) Use a graphing utility to graph each model separately with the data in the same viewing window. How well does each model fit the data? (d) Use each model to predict the year in which the revenue per share is about \(\$ 37\). Explain any differences in the predictions.

You divide a polynomial by another polynomial. The remainder is zero. What conclusion(s) can you make?

Price of Gold The table shows the average annual prices \(P\) (in dollars) of gold for the years 1996 to \(2005 .\) (Source: World Gold Council) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1996 & 1997 & 1998 & 1999 \\ \hline \text { Price of gold, } P & 387.82 & 330.98 & 294.12 & 278.55 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 \\ \hline \text { Price of gold, } P & 279.10 & 272.67 & 309.66 & 362.91 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|} \hline \text { Year } & 2004 & 2005 \\ \hline \text { Price of gold, } P & 409.17 & 444.47 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to 1996 . (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model from part (b) in the same viewing window as the scatter plot. (d) Use the graph of the model from part (c) to estimate when the price of gold was the lowest. Does this result agree with the actual data?