Chapter 2

A problem situation is given. (a) Decide what is being asked for, and label the unknown quantities. (b) Translate the verbal statements in the problem and the relationships between the known and unknown quantities into mathematical language, using a table as in Examples \(1-3\) (pages \(101 103\) ). The table is provided in Exercises 1 and \(2 .\) You need not find an equation to be solved. A student has exam scores of \(88,62,\) and \(79 .\) What score does he need on the fourth exam to have an average of \(80 ?\) $$\begin{array}{l|c}\text { English Language } & \begin{array}{c}\text { Mathematical } \\\\\text { Language }\end{array} \\\\\hline \text { Score on fourth exam } & \\\\\text { Sum of scores on four exams } & \\\\\text { Average of scores on four exams } &\end{array}$$

Find the coordinates of the highest or lowest point on the part of the graph of the equation in the given viewing window. Only the range of \(x\) -coordinates for the window are given \(_{i}\) you must choose an appropriate range of \(y\) -coordinates. \(y=2 x^{3}-3 x^{2}-12 x+1 ; \quad\) highest point when \(-3 \leq x \leq 3\)

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=|x-2|$$

A problem situation is given. (a) Decide what is being asked for, and label the unknown quantities. (b) Translate the verbal statements in the problem and the relationships between the known and unknown quantities into mathematical language, using a table as in Examples \(1-3\) (pages \(101 103\) ). The table is provided in Exercises 1 and \(2 .\) You need not find an equation to be solved. How many gallons of a \(12 \%\) salt solution should be combined with 10 gallons of an \(18 \%\) salt solution to obtain a \(16 \%\) solution? $$\begin{array}{l|c}\text { English Language } & \begin{array}{c}\text { Mathematical } \\\\\text { Language }\end{array} \\\\\hline \text { Gallons of 12\% solution } & \\\\\text { Total gallons of mixture } & \\\\\text { Amount of salt in 10 gallons of the } & \\\18 \% \text { solution } & \\\\\text { Amount of salt in the 12\% solution } & \\\\\text { Amount of salt in the mixture } &\end{array}$$

Find the coordinates of the highest or lowest point on the part of the graph of the equation in the given viewing window. Only the range of \(x\) -coordinates for the window are given \(_{i}\) you must choose an appropriate range of \(y\) -coordinates. $$\begin{aligned} &y=2 x^{6}+3 x^{5}+3 x^{3}-2 x^{2} ; \quad \text { lowest point when }\\\ &-3 \leq x \leq 3 \end{aligned}$$

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=\sqrt{x+5}$$

Find the coordinates of the highest or lowest point on the part of the graph of the equation in the given viewing window. Only the range of \(x\) -coordinates for the window are given \(_{i}\) you must choose an appropriate range of \(y\) -coordinates. $$y=\frac{4}{x^{2}}-\frac{7}{x}+1 ; \quad \text { lowest point when }-10 \leq x \leq 10$$

Determine graphically the number of solutions of the equation, but don't solve the equation. You may need a viewing window other than the standard one to find all the \(x\) -intercepts. $$x^{7}-10 x^{5}+15 x+10=0$$

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=x^{2}-x$$

A problem situation is given. (a) Decide what is being asked for, and label the unknown quantities. (b) Translate the verbal statements in the problem and the relationships between the known and unknown quantities into mathematical language, using a table as in Examples \(1-3\) (pages \(101 103\) ). The table is provided in Exercises 1 and \(2 .\) You need not find an equation to be solved. A triangle has area 96 square inches, and its height is twothirds of its base. What are the base and height of the triangle?

Find the coordinates of the highest or lowest point on the part of the graph of the equation in the given viewing window. Only the range of \(x\) -coordinates for the window are given \(_{i}\) you must choose an appropriate range of \(y\) -coordinates. $$y=\frac{1}{x^{2}+2 x+2} ; \quad \text { highest point when }-5 \leq x \leq 5$$

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=x^{2}+x+1$$

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=x^{3}+1$$

Find the coordinates of the highest or lowest point on the part of the graph of the equation in the given viewing window. Only the range of \(x\) -coordinates for the window are given \(_{i}\) you must choose an appropriate range of \(y\) -coordinates. $$y=\frac{x^{2}(x+1)^{3}}{(x-2)(x-4)^{2}} ; \quad \text { lowest point when }-10 \leq x \leq-1$$

Determine graphically the number of solutions of the equation, but don't solve the equation. You may need a viewing window other than the standard one to find all the \(x\) -intercepts. $$6 x^{5}+80 x^{3}+45 x^{2}+30=45 x^{4}+86 x$$

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=\frac{1}{x}$$

Find the highest point on the part of the graph of \(y=x^{3}-3 x+2\) that is shown in the given window. The answers are not all the same. (a) \(-2 \leq x \leq 0\) (b) \(-2 \leq x \leq 2\) (c) \(-2 \leq x \leq 3\)

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{3}+4 x^{2}+10 x+15=0 ; \quad(-3,-2)$$

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$3+y=.5 x$$

Set up the problem by labeling the unknowns, translating the given information into mathematical language, and finding an equation that will produce the solution to the problem. You need not solve this equation. A corner lot has dimensions 25 by 40 yards. The city plans to take a strip of uniform width along the two sides bordering the streets to widen these roads. How wide should the strip be if the remainder of the lot is to have an area of 844 square yards?

Find the lowest point on the part of the graph of \(y=x^{3}-3 x+2\) that is shown in the given window. (a) \(0 \leq x \leq 2\) (b) \(-2 \leq x \leq 2\) (c) \(-3 \leq x \leq 2\)

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{3}+9=3 x^{2}+6 x$$

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$y-2 x=4$$

In the remaining exercises, solve the applied problems. You have already invested $$ 550\( in a stock with an annual return of \)11 \% .\( How much of an additional $$ 1100 should be invested at \)12 \%\( and how much at \)6 \%\( so that the total return on the entire 1650 dollars is \)9 \% ?$

The fuel economy \(y\) of a representative car (in miles per gallon ) can be approximated by \(y=-.00000636 x^{4}+.001032 x^{3}-.067 x^{2}+2.19 x+8.6\) where \(x\) is the speed of the car (in miles per hour)." At what speed does this car get the most miles per gallon?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{4}+x-3=0 ; \quad(-\infty, 0)$$

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$y=x^{2}-5 x+2$$

In the remaining exercises, solve the applied problems. If you borrow $$ 500\( from a credit union at \)12 \%\( annual interest and \)\$ 250\( from a bank at \)18 \%\( annual interest, what is the effective annual interest rate (that is, what single rate of interest on $$ 750\) would result in the same total amount of interest)?

Between 1997 and \(2005,\) the number \(y\) of unemployed (in thousands) was approximated by $$ \begin{array}{r} y=-53.4 x^{3}+1772.33 x^{2}-18,681.32 x+69,188.1 \\ (7 \leq x \leq 15) \end{array} $$ where \(x=7\) corresponds to \(1997 .^{\\#}\) In what year was unemployment the highest?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{5}+5=3 x^{4}+x ; \quad(2, \infty)$$

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$y=3 x^{2}+x-4$$

In the remaining exercises, solve the applied problems. A radiator contains 8 quarts of fluid, \(40 \%\) of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the new mixture is \(60 \%\) antifreeze?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$\sqrt{x^{4}+x^{3}-x-3}=0 ; \quad(-\infty, 0)$$

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$y=2 x^{3}+1 x^{2}-4 x+1$$

In the remaining exercises, solve the applied problems. A radiator contains 10 quarts of fluid, \(30 \%\) of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the new mixture is \(40 \%\) antifreeze?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$\sqrt{8 x^{4}-14 x^{3}-9 x^{2}+11 x-1}=0 ; \quad(-\infty, 0)$$

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$y=2 x^{4}-2 x^{3}-2 x^{2}-2 x+5$$

In the remaining exercises, solve the applied problems. An airplane flew with the wind for 2.5 hours and returned the same distance against the wind in 3.5 hours. If the cruising speed of the plane was a constant \(360 \mathrm{mph}\) in air, how fast was the wind blowing?

A farmer has 1800 feet of fencing. He plans to enclose a rectangular region bordering a river (with no fencing needed along the river side). What dimensions should he use to have an enclosure of largest possible area?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$\sqrt{\frac{2}{5} x^{5}+x^{2}-2 x}=0 ; \quad(0, \infty)$$

(a) Graph $$ y=\frac{1}{x^{2}+1} $$ in the standard window. (b) Does the graph appear to stop abruptly partway along the \(x\) -axis? Use the trace feature to explain why this happens. [Hint: In this viewing window, each pixel represents a rectangle that is approximately .32 unit high.] (c) Find a viewing window with \(-10 \leq x \leq 10\) that shows a complete graph that does not fade into the \(x\) -axis.

In the remaining exercises, solve the applied problems. A train leaves New York for Boston, 200 miles away, at 3: 00 P.M. and averages 75 mph. Another train leaves Boston for New York on an adjacent set of tracks at 5: 00 P.M. and averages 45 mph. At what time will the trains meet?

A rectangular field will be fenced on all four sides. Fencing for the north and south sides costs \(\$ 5\) per foot and fencing for the other two sides costs \(\$ 10\) per foot. What is the maximum area that can be enclosed for \(\$ 5000 ?\)

(a) Graph \(y=x^{3}-2 x^{2}+x-2\) in the standard window. (b) Use the trace feature to show that the portion of the graph with \(0 \leq x \leq 1.5\) is not actually horizontal. \([\) Hint: All the points on a horizontal segment must have the same \(y\) -coordinate (why?).] (c) Find a viewing window that clearly shows that the graph is not horizontal when \(0 \leq x \leq 1.5\)

In the remaining exercises, solve the applied problems. The average of two real numbers is \(41.125,\) and their product is \(1683 .\) What are the numbers? [Hint: See Example \(1 .]\)

A rectangular area of 24,200 square feet is to be fenced on all four sides. Fencing for the east and west sides costs \(\$ 10\) per foot and fencing for the other two sides costs \(\$ 20\) per foot. What is the cost of the least expensive fence?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{2}=\sqrt{x+5} ; \quad(-2,-1)$$

In the remaining exercises, solve the applied problems. A rectangle is twice as long as it is wide. If it has an area of 24.5 square inches, what are its dimensions? [Hint: See Example \(2 .]\)

A fence is needed to enclose an area of 30,246 square feet. One side of the area is bounded by an existing fence, so no new fencing is needed there. Fencing for the side opposite the existing fence costs \(\$ 18\) per foot. Fencing for the other two sides costs \(\$ 6\) per foot. What is the cost of the least expensive fence?

In Exercises \(15-24,\) use the techniques of Examples 4 and 5 to graph the equation in a suitable square viewing window. $$y^{2}=x-2$$