Chapter 8

Use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) Cost \(C=2.65 x+350,000\) Revenue \(R=4.15 x\)

Write the partial fraction decomposition for the rational function. Identify the graph of the rational function and the graph of each term of its decomposition. State any relationship between the vertical asymptotes of the rational function and the vertical asymptotes of the terms of the decomposition. $$y=\frac{2(4 x-3)}{x^{2}-9}$$

$$\left\\{\begin{aligned} 2 x+y-z+2 w &=-6 \\ 3 x+4 y &+w=1 \\ x+5 y+2 z+6 w &=-3 \\ 5 x+2 y-z-w &=3 \end{aligned}\right.$$

Solve the equation algebraically. Round your result to three decimal places. $$e^{2 x}-10 e^{x}+24=0$$

An airplane flying into a headwind travels the 1800 -mile flying distance between New York City and Albuquerque, New Mexico, in 3 hours and 36 minutes. On the return flight, the same distance is traveled in 3 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant

Use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) Cost \(C=5.5 \sqrt{x}+10,000\) Revenue \(R=3.29 x\)

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$c A-B C$$

An object moving vertically is at the given heights at the specified times. Find the position equation \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\) for the object. At \(t=1\) second, \(s=128\) feet. At \(t=2\) seconds, \(s=80\) feet. At \(t=3\) seconds, \(s=0\) feet.

Determine whether the two systems of linear equations yield the same solution. If so, find the solution. $$\left\\{\begin{aligned} x-2 y+z &=-6 \\ y-5 z &=16 \\ z &=-3 \end{aligned} \quad\left\\{\begin{array}{r} x+y-2 z=6 \\ y+3 z=-8 \\ z=-3 \end{array}\right.\right.$$

A property of determinants is given \((A \text { and } B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtaincd from \(A\) by interchanging two rows of \(A\) or by interchanging two columns of \(A,\) then \(|B|=-|A|\) $$\text { (a) }\left|\begin{array}{rrr}1 & 3 & 4 \\\\-7 & 2 & -5 \\\6 & 1 & 2\end{array}\right|=-\left|\begin{array}{rrr}1 & 4 & 3 \\\\-7 & -5 & 2 \\\6 & 2 & 1\end{array}\right|$$ $$\text { (b) }\left|\begin{array}{rrr}1 & 3 & 4 \\\\-2 & 2 & 0 \\\1 & 6 & 2\end{array}\right|=-\left|\begin{array}{rrr}1 & 6 & 2 \\\\-2 & 2 & 0 \\\1 & 3 & 4\end{array}\right|$$

Solve the equation algebraically. Round your result to three decimal places. $$7 \ln 3 x=12$$

Navigation A motorboat traveling with the current takes 40 minutes to travel 20 miles downstream. The return trip takes 60 minutes. Find the speed of the current and the speed of the boat relative to the current, assuming that both remain constant.

Use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) Cost \(C=7.8 \sqrt{x}+18,500\) Revenue \(R=12.84 x\)

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$d A B^{2}$$

An object moving vertically is at the given heights at the specified times. Find the position equation \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\) for the object. At \(t=1\) second, \(s=32\) feet. At \(t=2\) seconds, \(s=32\) feet. At \(t=3\) seconds, \(s=0\) feet.

A property of determinants is given \((A \text { and } B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by adding a multiple of a row of \(A\) to another row of \(A\) or by adding a multiple of a column of \(A\) to another column of \(A,\) then \(|B|=|A|\) $$\begin{array}{l|lr} \text { (a) }\left|\begin{array}{ll} 1 & -3 \\ 5 & 2 \end{array}\right| & =\left|\begin{array}{ll} 1 & -3 \\ 0 & 17 \end{array}\right| \end{array}$$ $$\text { (b) }\left|\begin{array}{rrr} 5 & 4 & 2 \\ 2 & -3 & 4 \\ 7 & 6 & 3 \end{array}\right|=\left|\begin{array}{rrr} 1 & 10 & -6 \\ 2 & -3 & 4 \\ 7 & 6 & 3 \end{array}\right|$$

Solve the equation algebraically. Round your result to three decimal places. $$\ln (x+9)=2$$

Business \(A\) minor league baseball team had a total attendance one evening of \(1175 .\) The tickets for adults and children sold for \(\$ 15\) and \(\$ 12,\) respectively. The ticket revenue that night was \(\$ 16,275\) (a) Create a system of linear equations to find the numbers of adults \(A\) and children \(C\) at the game. (b) Solve your system of equations by elimination or by substitution. Explain your choice. (c) Use the intersect feature or the zoom and trace features of a graphing utility to solve your system.

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 30 meters and the length is 3 meters greater than the width.

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$c d A+B$$

An object moving vertically is at the given heights at the specified times. Find the position equation \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\) for the object. At \(t=1\) second, \(s=352\) feet. At \(t=2\) seconds, \(s=272\) feet. At \(t=3\) seconds, \(s=160\) feet.

Determine whether the two systems of linear equations yield the same solution. If so, find the solution. $$\left\\{\begin{aligned} x-4 y+5 z &=27 \\ y-7 z &=-54 \\ z &=8 \end{aligned}\left\\{\begin{aligned} x-6 y+z &=15 \\ y+5 z &=42 \\ z &=8 \end{aligned}\right.\right.$$

A property of determinants is given \((A \text { and } B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by multiplying a row of \(A\) by a nonzero constant \(c\) or by multiplying a column of \(A\) by a nonzero constant \(c,\) then \(|B|=c|A|\) $$\begin{array}{l|ll} \text { (a) } & \begin{array}{ll} 1 & 5 \\ 6 & 9 \end{array}|=3| \begin{array}{ll} 1 & 5 \\ 2 & 3 \end{array} | \end{array}$$ $$\text { (b) }\left|\begin{array}{ll} 2 & 8 \\ 6 & 8 \end{array}\right|=8\left|\begin{array}{ll} 1 & 2 \\ 3 & 2 \end{array}\right|$$

Chemistry Thirty liters of a \(40 \%\) acid solution are obtained by mixing a \(25 \%\) solution with a \(50 \%\) solution. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the percent of acid in the final mixture. Let \(x\) and \(y\) represent the amounts of the \(25 \%\) and \(50 \%\) solutions, respectively. (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. (c) As the amount of the \(25 \%\) solution increases, how does the amount of the \(50 \%\) solution change? (d) How much of each solution is required to obtain the specified concentration of the final mixture?

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 280 centimeters and the width is 20 centimeters less than the length.

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$c A+d B$$

An object moving vertically is at the given heights at the specified times. Find the position equation \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\) for the object. At \(t=1\) second, \(s=132\) feet. At \(t=2\) seconds, \(s=100\) feet. At \(t=3\) seconds, \(s=36\) feet.

Determine whether the two systems of linear equations yield the same solution. If so, find the solution. $$\left\\{\begin{aligned} x+3 y-z=& 19 \\ y+6 z=&-18 \\ z=&-4 \end{aligned}\left\\{\begin{array}{rr} x-y+3 z= & -21 \\ y-2 z= & 14 \\ z= & -4 \end{array}\right.\right.$$

Exploration A diagonal matrix is a square matrix with all zero entries above and below its main diagonal. Evaluate the determinant of each diagonal matrix. Make a conjecture based on your results. $$\text { (a) }\left[\begin{array}{ll} 7 & 0 \\ 0 & 4 \end{array}\right]$$ $$\text { (b) }\left[\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{array}\right]$$ $$(c)\left[\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right]$$

Business A grocer sells oranges for \(\$ 0.95 each and grapefruits for \)\$ 1.05\( each. You purchased a mix of 16 oranges and grapefruits and paid \$ 15.90 .\) How many of each type of fruit did you buy?

The daily DVD rentals of a newly released animated film and a newly released horror film from a movie rental store can be modeled by the equations $$\left\\{\begin{array}{ll} N=360-24 x & \text { Animated film } \\ N=24+18 x & \text { Horror film } \end{array}\right.$$ where \(N\) is the number of DVDs rented and \(x\) represents the week, with \(x=1\) corresponding to the first week of release. (a) Use the table feature of a graphing utility to find the numbers of rentals of each movie for each of the first 12 weeks of release. (b) Use the results of part (a) to determine the solution to the system of equations. (c) Solve the system of equations algebraically. (d) Compare your results from parts (b) and (c). (e) Interpret the results in the context of the situation.

Operations with Matrices Use a graphing utility to perform the indicated operations. \(A=\left[\begin{array}{ll}2 & 0 \\ 4 & 5\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{ll}5 & 3 \\ 1 & 4\end{array}\right]\) $$A^{2}-5 A+2 I_{2}$$

Find the equation of the parabola $$y=a x^{2}+b x+c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(0,0),(3,0),(4,4)$$

Nutrition Two cheeseburgers and one small order of french fries from a fast- food restaurant contain a total of 830 calories. Three cheeseburgers and two small orders of french fries contain a total of 1360 calories. Find the number of calories in each item.

You want to buy either a wood pellet stove or an electric furnace. The pellet stove costs \(\$ 3650\) and produces heat at a cost of \(\$ 19.15\) per 1 million Btu (British thermal units). The electric furnace costs \(\$ 2780\) and produces heat at a cost of \(\$ 33.25\) per 1 million Btu. (a) Write a function for the total cost \(y\) of buying the pellet stove and producing \(x\) million Btu of heat. (b) Write a function for the total cost \(y\) of buying the electric furnace and producing \(x\) million Btu of heat. (c) Use a graphing utility to graph and solve the system of equations formed by the two cost functions. (d) Solve the system of equations algebraically. (e) Interpret the results in the context of the situation.

Operations with Matrices Use a graphing utility to perform the indicated operations. \(A=\left[\begin{array}{ll}2 & 0 \\ 4 & 5\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{ll}5 & 3 \\ 1 & 4\end{array}\right]\) $$B^{2}-7 B+6 I_{2}$$

Find the equation of the parabola $$y=a x^{2}+b x+c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(0,5),(1,6),(2,5)$$

Explain why the determinant of the matrix is equal to zero. $$\left[\begin{array}{rrr} 2 & -4 & 5 \\ 1 & -2 & 3 \\ 0 & 0 & 0 \end{array}\right]$$

Sales The projected sales \(S\) (in millions of dollars) of two clothing retailers from 2015 through 2020 can be modeled by \(\left\\{\begin{array}{ll}S-149.9 t=415.5 & \text { Retailer } \mathrm{A} \\\ S-183.1 t=117.3 & \text { Retailer } \mathrm{B}\end{array}\right.\) where \(t\) is the year, with \(t=5\) corresponding to 2015 (a) Solve the system of equations using the method of your choice. Explain why you chose that method. (b) Interpret the meaning of the solution in the context of the problem. (c) Interpret the meaning of the coefficient of the \(t\) -term in each model. (d) Suppose the coefficients of \(t\) were equal and the models remained the same otherwise. How would this affect your answers in parts (a) and (b)?

A small software company invests \(\$ 16,000\) to produce a software package that will sell for \(\$ 55.95 .\) Each unit can be produced for \(\$ 9.45\) (a) Write the cost and revenue functions for \(x\) units produced and sold. (b) Use a graphing utility to graph the cost and revenue functions in the same viewing window. Use the graph to approximate the number of units that must be sold to break even and verify the result algebraically.

Find the equation of the parabola $$y=a x^{2}+b x+c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(-1,1),(0,-4),(1,-13)$$

You are offered two jobs selling college textbooks. One company offers an annual salary of \(\$ 33,000\) plus a year-end bonus of \(1 \%\) of your total sales. The other company offers an annual salary of \(\$ 30,000\) plus a year- end bonus of \(2.5 \%\) of your total sales. How much would you have to sell in a year to make the second offer the better offer?

Find the equation of the parabola $$y=a x^{2}+b x+c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(-2,9),(-1,0),(1,6)$$

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{array}{l} 5 b+10 a=20.2 \\ 10 b+30 a=50.1 \end{array}\right.$$

What are the dimensions of a rectangular tract of land with a perimeter of 40 miles and an area of 96 square miles?

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(0,0),(5,5),(10,0)$$

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the equations. Solve the system using matrices. $$\begin{aligned} &f(-2)=-15\\\ &f(-1)=7\\\ &f(1)=-3 \end{aligned}$$

Factor the expression. $$4 y^{2}-12 y+9$$

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{aligned} 5 b+10 a &=11.7 \\ 10 b+30 a &=25.6 \end{aligned}\right.$$

What are the dimensions of an isosceles right triangle with a two-inch hypotenuse and an area of 1 square inch?