Chapter 12.4, Problem 42E

a.

To determine

To find: A model for the average cost per ton of material recycled.

Answer to Problem 42E

  $\text{Average cost per unit}$

  $\overline{C}=1.25+\frac{10500}{x}.$

Explanation of Solution

Given information: The cost function for a company recycle x tons material is

  $C=1.25x+10,500,$ where C is in dollars.

Calculation:

To find average cost/unit.

Average cost per unit= $\frac{Total\cos t}{no.ofunits}$ .

  $\begin{array}{l}\text{Average cost per unit}=\frac{1.25x+10500}{x}.\\ \overline{C}=1.25+\frac{10500}{x}.\end{array}$

b.

To determine

To find: The average costs of recycling 100,1000, and 10,000tons of material.

Answer to Problem 42E

  $\begin{array}{l}x=100,\overline{C}=106.25\$.\\ x=1000,\overline{C}=11.75\$.\\ x=10,000,\overline{C}=1.30\$.\end{array}$

Explanation of Solution

Given information: The cost function for a company recycle x tons material is

  $C=1.25x+10,500,$ where C is in dollars.

Calculation:

  $\text{Average cost per unit}$

  $\begin{array}{l}\overline{C}=1.25+\frac{10500}{x}\mathrm{..}\\ x=100.\\ \overline{C}=1.25+\frac{10500}{100}=106.25\$\\ x=1000.\\ \overline{C}=1.25+\frac{10500}{1000}.=11.75\$\\ x=10,000.\\ \overline{C}=1.25+\frac{10500}{10000}.=1.30\$\end{array}$

c.

To determine

To find: The limit of the average cost function as x approaches infinity .Explain the meaning of the limit in the context of the problem.

Answer to Problem 42E

  $\underset{\delta x\to \infty }{\lim }\overline{C}=1.25\$.$

It means if ,large amount of material is recycled the average cost per ton of material recycled approached % 1.25.

Explanation of Solution

Given information: The cost function for a company recycle x tons material is

  $C=1.25x+10,500,$ where C is in dollars.

Calculation:

  $\begin{array}{l}\underset{\delta x\to \infty }{\lim }\overline{C}=\underset{\delta x\to \infty }{\lim }(1.25+\frac{10500}{x})=\underset{\delta x\to \infty }{\lim }1.25+\underset{\delta x\to \infty }{\lim }\frac{10500}{x}.\\ \underset{\delta x\to \infty }{\lim }\overline{C}=1.25+0=\mathrm{1.25.}\\ \underset{\delta x\to \infty }{\lim }\overline{C}=1.25\$.\end{array}$

It means if ,large amount of material is recycled the average cost per ton of material recycled approached % 1.25.