Chapter 6

Solve each application. A company produces two models of bicycles: model \(A\) and model \(B\). Model \(A\) requires 2 hours of assembly time, and model \(B\) requires 3 hours of assembly time. The parts for model \(A\) cost \(\$ 25\) per bike; those for model \(B\) cost \(\$ 30\) per bike. If the company has a total of 34 hours of assembly time and \(\$ 365\) available per day for these two models, what is the maximum number of each model that can be made in a day and use all of the available resources?

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$B A$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned} &\begin{array}{l}5 x-y=-4 \\\3 x+2 z=4\end{array}\\\&4 y+3 z=22\end{aligned}$$

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$y \leq x^{2}+5$$

Solve each nonlinear system of equations analytically. $$\begin{array}{r}x^{2}+y^{2}=5 \\\\-3 x+4 y=2\end{array}$$

Solve each application. Scheduling Production Caltek Computer Company makes two products: computer monitors and printers. Both require time on two machines: monitors, 1 hour on machine \(A\) and 2 hours on machine \(B\); printers, 3 hours on machine \(A\) and 1 hour on machine \(B\). Both machines operate 15 hours per day. What is the maximum number of each product that can be produced per day under these conditions?

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$A C$$

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities.$$\begin{aligned} &x+y \geq 2\\\ &x+y \leq 6 \end{aligned}$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}x^{2}+y^{2} &=10 \\\\-x^{2}+y &=-4\end{aligned}$$

Solve each application. Financing Expansion To get funds necessary for a planned expansion, a small company took out three loans totaling \(\$ 12,500 .\) The company was able to borrow some of the money at \(2 \% .\) It borrowed \(\$ 1000\) more than \(\frac{1}{2}\) the amount of the \(2 \%\) loan at \(3 \%\) and the rest at \(2.5 \% .\) The total annual interest was \(\$ 305 .\) How much did the company borrow at each rate?

Position of a Particle Suppose that the position of a particle moving along a straight line is given by $$s(t)=a t^{2}+b t+c$$ where \(t\) is time in seconds and \(a, b,\) and \(c\) are real numbers. If \(s(0)=5, s(1)=23,\) and \(s(2)=37,\) find the equation that defines \(s(t) .\) Then find \(s(8)\)

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$B C$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{r}2 x-y+3 z=1 \\\\-2 x+y-3 z=2 \\\5 x-y+z=2\end{array}$$

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned} &y \geq|x+2|\\\ &y \leq 6 \end{aligned}$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=|x-1|\\\&y=x^{2}-4\end{aligned}$$

Position of a Particle Suppose that the position of a particle moving along a straight line is given by $$s(t)=a t^{2}+b t+c$$ where \(t\) is time in seconds and \(a, b,\) and \(c\) are real numbers. If \(s(0)=-10, s(1)=6,\) and \(s(2)=30,\) find the equation that defines \(s(t) .\) Then find \(s(10)\)

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$C B$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}-2 x-2 y+3 z &=4 \\\5 x+7 y-z &=2 \\\2 x+2 y-3 z &=-4\end{aligned}$$

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned} &y \geq 2^{x}\\\ &y \leq 8 \end{aligned}$$

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\log (x+5)\\\&y=x^{2}\end{aligned}$$

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$A B$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}3 x+2 y-w &=0 \\\2 x+z+2 w &=5 \\\x+2 y-z &=-2 \\\2 x-y+z+w &=2\end{aligned}$$

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$y \leq x^{3}+x^{2}-4 x-4$$

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}y &=5^{x} \\\x y &=1\end{aligned}$$

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$C A$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}x+2 y-z+w &=8 \\\2 x-y+2 w &=8 \\\y+3 z &=5 \\\x-z &=4\end{aligned}$$

The table lists annual enrollment \(y\) in thousands for the Head Start program \(x\) years after 1980 . $$\begin{array}{|c|c|c|c|} \hline x & 0 & 10 & 32 \\ \hline y & 376 & 541 & 1128 \\ \hline \end{array}$$

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned} &y=2 x-1\\\ &y=2-x^{2} \end{aligned}$$

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=e^{x+1}\\\&2 x+y=3\end{aligned}$$

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$A^{2}$$

In your own words, explain what happens when you apply Cramer's rule if \(D=0\)

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned} &y=x^{2}-x+1\\\ &y=-x^{2}+1 \end{aligned}$$

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\sqrt[3]{x-4}\\\&x^{2}+y^{2}=6\end{aligned}$$

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$A^{3}$$

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned} &y=x^{3}\\\ &y=x \end{aligned}$$

The total spending on Black Friday during 2011 and 2012 was 1858 million dollars. From 2011 to \(2012,\) spending increased by 226 million dollars. (Source: www.marketingcharts.com) (a) Write a system of equations whose solution represents the Black Friday spending in each of these years. Let \(x\) be the amount spent in 2012 and \(y\) be the amount spent in 2011. (b) Solve the system. (c) Interpret the solution.

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned} &y=2 x^{2}+x-3\\\ &y=x^{2}-2 x+1 \end{aligned}$$

The average of self-reported spending "yesterday" for high-income consumers and middle-/low-income consumers was 93.50 dollars in September 2012 . High- income consumers spend 65 dollars more than middle-/low-income consumers. (Source: www.marketingcharts.com) (a) Write a system of equations whose solution gives the self-reported spending for each income group. Let \(x\) be the spending by high-income consumers and \(y\) be the spending by middle-/low-income consumers. (b) Solve the system. (c) Interpret the solution.

For any matrices \(P\) and \(Q,\) what must be true for both \(P Q\) and \(Q P\) to exist?

Fawn Population To model spring fawn count \(F\) from adult antelope population \(A\), precipitation \(P\), and severity of winter \(W,\) environmentalists have used the equation. $$F=a+b A+c P+d W$$ where the coefficients \(a, b, c,\) and \(d\) are constants that must be determined before using the equation. The table lists the results of four different (representative) years. $$\begin{array}{|c|c|c|c|} \text { Fawns } & \text { Adults } & \begin{array}{c} \text { Precip. } \\ \text { (in inches) } \end{array} & \begin{array}{c} \text { Winter } \\ \text { Severity } \end{array} \\ \hline 239 & 871 & 11.5 & 3 \\ 234 & 847 & 12.2 & 2 \\ 192 & 685 & 10.6 & 5 \\ 343 & 969 & 14.2 & 1 \end{array}$$ A. Substitute the values for \(F, A, P,\) and \(W\) from the table for each of the four years into the given equation \(F=a+b A+c P+d W\) to obtain four linear equations involving \(a, b, c,\) and \(d\) B. Write a \(4 \times 5\) augmented matrix representing the system, and solve for \(a, b, c,\) and \(d\) C. Write the equation for \(F\), using the values from part (b) for the coefficients. D. If a winter has severity \(3,\) adult antelope population 960 and precipitation 12.6 inches, predict the spring fawn count. (Compare this with the actual count of \(320 .\) )

From 2010 to \(2012,\) the average selling price of tablets decreased by \(30 \% .\) This percent reduction amounted in a decrease of 195 dollars. Find the average selling price of tablets in 2010 and in 2012.

Find a system of linear inequalities for which the graph is the region in the first quadrant between and inclusive of the pair of lines \(x+2 y-8=0\) and \(x+2 y=12\)

Yagel's Yogurt sells three types of yogurt: nonfat, regular, and supercreamy, at three locations. Location I sells 50 gallons of nonfat, 100 gallons of regular, and 30 gallons of supercreamy each day. Location II sells 10 gallons of nonfat, 90 gallons of regular, and 50 gallons of supercreamy each day. Location III sells 60 gallons of nonfat, 120 gallons of regular, and 40 gallons of supercreamy each day. (a) Write a \(3 \times 3\) matrix that shows sales for the three locations, with the rows representing the locations. (b) The incomes per gallon for nonfat, regular, and supercreamy are \(\$ 12, \$ 10,\) and \(\$ 15,\) respectively. Write a \(3 \times 1\) matrix displaying the incomes per gallon. (c) Find a matrix product that gives the daily income at each of the three locations. (d) What is Yagel's Yogurt's total daily income from the three locations?

A box has an open top, rectangular sides, and a square base. The volume of the box is 576 cubic inches, and the surface area of the outside of the box is 336 square inches. Find the dimensions of the box.

The table shows weight \(W,\) neck size \(N,\) overall length \(L,\) and chest size \(C\) for four bears. $$\begin{array}{|c|c|c|c|} \hline W \text { (pounds) } & N \text { (inches) } & L \text { (inches) } & C \text { (inches) } \\ \hline 125 & 19 & 57.5 & 32 \\\ 316 & 26 & 65 & 42 \\ 436 & 30 & 72 & 48 \\ 514 & 30.5 & 75 & 54 \end{array}$$ A. We can model these data with the equation $$ W=a+b N+c L+d C $$ where \(a, b, c,\) and \(d\) are constants. To do so, represent a system of linear equations by a \(4 \times 5\) augmented matrix whose solution gives values for \(a, b, c,\) and \(d\) B. Solve the system. Round each value to the nearest thousandth. C. Predict the weight of a bear with \(N=24, L=63\) and \(C=39 .\) Interpret the result.

A box has rectangular sides and a rectangular top and base that are twice as long as they are wide. The volume of the box is 588 cubic inches, and the surface area of the outside of the box is 448 square inches. Find the dimensions of the box.

In certain parts of the Rocky Mountains, deer are the main food source for mountain lions. When the deer population \(d\) is large, the mountain lions ( \(m\) ) thrive. However, a large mountain lion population drives down the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$\left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{cc} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 year? 2 years? (c) Consider part (b), but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of \(1 \$ 6\).

A student invests a total of 5000 dollars at \(3 \%\) and \(4 \%\) annually. After 1 year, the student receives a total of 187.50 dollars in interest. How much did the student invest at each interest rate?

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

In one study, a group of conditioned athletes was exercised to exhaustion. Let \(x\) represent an athlete's heart rate 5 seconds after stopping exercise and \(y\) the rate after 10 seconds. It was found that the maximum heart rate \(H\) for these athletes satisfied the two equations $$\begin{aligned}&H=0.491 x+0.468 y+11.2\\\&H=-0.981 x+1.872 y+26.4\end{aligned}$$ and If an athlete had maximum heart rate \(H=180\), determine \(x\) and \(y\) graphically. Interpret your answer. (Source: Thomas, V., Science and Sport, Faber and Faber.)