Chapter 1.10, Problem 76E

a.

To determine

Assuming that the relationship between cost and number of chairs produced is linear, find an equation that expresses this relationship. Then graph the equation.

Answer to Problem 76E

  $C=13n+900$

  

Explanation of Solution

Calculation:

Assume that the relationship between cost and number of chairs produced is linear, and then the equation that represents this relationship is given as,

  $C=mn+b$

To manufacture $100$ chairs, it costs $\$2200$ . So, substitute $n=100$ and $C=2200$ in the equation $C=mn+b$ , we get,

  $\begin{array}{l}2200=m\left(100\right)+b\\ 2200=100m+b\mathrm{............}\left(1\right)\end{array}$

To manufacture $300$ chairs, it costs $\$4800$ . So, substitute $n=300$ and $C=4800$ in the equation $C=mn+b$ , we get,

  $\begin{array}{l}4800=m\left(300\right)+b\\ 4800=300m+b\mathrm{............}\left(2\right)\end{array}$

Subtract equation $\left(2\right)$ from equation $\left(1\right)$ ,

  $\begin{array}{l}\begin{array}{c}4800=300m+b\\ -2200=-100m-b\\ 2600=200m\end{array}\\ 13=m\\ m=13\end{array}$

So, the value of $m$ is $m=13$ .

Now, we substitute the value $m=13$ in equation $\left(1\right)$ , we get:

  $\begin{array}{l}2200=100\left(13\right)+b\\ 2200=1300+b\\ b=900\end{array}$

Put these values of $m=13$ and $b=900$ in $C=mn+b$ to get,

  $C=13n+900$

So, the linear equation that expresses this relationship between cost and number of chairs produced is $C=13n+900$ .

The graph of the equation $C=13n+900$ is given below,

  

Hence, the linear equation that expresses this relationship between cost and number of chairs produced is $C=13n+900$ and the graph is shown above.

b.

To determine

What is the slope of the line in part $\left(a\right)$ and what does it represent?

Answer to Problem 76E

  $13$

It represents the cost per number of chairs.

Explanation of Solution

Calculation:

Since, the equation of the line that has slope $m$ and $y-\text{intercept}$$b$ is $y=mx+b$ .

Hence, the slope of the line $C=13n+900$ is $13$ and it represents the cost per number of chairs.

c.

To determine

What is $y-\text{intercept}$ of this line, what does it represent?

Answer to Problem 76E

  $900$

It represents the monthly fixed cost.

Explanation of Solution

Calculation:

Since, the equation of the line that has slope $m$ and $y-\text{intercept}$$b$ is $y=mx+b$ .

Hence, the $y-\text{intercept}$ of the line $C=13n+900$ is $900$ and it represents the monthly fixed cost.