Chapter 11.4, Problem 58E

(a)

To determine

To calculate:

Model for average cost per ton of organic material recycled.

Answer to Problem 58E

Model for the average cost per ton of organic recycle is $\overline{C}=\frac{50x+2700}{x}$

Explanation of Solution

Given information:

  $C=50x+2700$

Where,

  $C=$ Cost in dollars.

Calculation:

Here, let’s assume $\overline{C}$ be the average cost per ton and $x$ be the amount of organic material to be recycle then,

  $\overline{C}=\frac{C}{x}$

But the cost of function for the new supermarket for $x$ ton organic material to be recycle is $C=50x+2700$

Then,

  $\begin{array}{l}\overline{C}=\frac{C}{x}\\ =\frac{50x+2700}{x}\end{array}$

The required model for the average cost per ton of organic recycle is $\overline{C}=\frac{50x+2700}{x}$

(b)

To determine

To calculate:

Average costs of recycling $100tons$ and $1000tons$ of organic material.

Answer to Problem 58E

Average cost of recycling $100tons$ and $1000tons$ of organic material is $\$77$

  $\$77$ and $\$50002.7$

Explanation of Solution

Given information:

  $C=50x+2700$

Where,

  $C=$ Cost in dollars.

Calculation:

For determining the average cost for $100tons$ and $1000tons$ of organic material.

Average cost $\overline{C}$ for $x$ tons to recycle is

  $\overline{C}=\frac{50x+2700}{x}$

So average cost for $100tons$ is:

  $\begin{array}{l}\overline{C}\left(100\right)=\frac{50\left(100\right)+2700}{100}\\ =\frac{5000+2700}{100}\\ =\frac{7700}{100}\\ =77\end{array}$

So average cost for $1000tons$ is:

  $\begin{array}{l}\overline{C}\left(1000\right)=\frac{50\left(1000\right)+2700}{1000}\\ =\frac{50000+2700}{1000}\\ =\frac{52700}{100}\\ =50002.7\end{array}$

Thus, required average cost of recycling $100tons$ and $1000tons$ of organic material is $\$77$ and $\$50002.7$

(c)

To determine

To calculate:

Limit of average cost function and explaining about the meaning of limit in context of the problem.

Answer to Problem 58E

Limit of the = average cost function is $\underset{x\to \infty }{\lim }\overline{C}=\$50$

Explanation of Solution

Given information:

  $C=50x+2700$

Where,

  $C=$ Cost in dollars.

Calculation:

To find out the Limit of average cost function,

Average cost function $\overline{C}$ for $x$ tons organic material to recycle is

  $\overline{C}=\frac{50x+2700}{x}$

Then, the limit of the average cost function as $x$ to infinity,

  $\begin{array}{l}\underset{x\to \infty }{\lim }\overline{C}\left(100\right)=\underset{x\to \infty }{\lim }\frac{50x+2700}{x}\\ =\underset{x\to \infty }{\lim }\frac{50x\left(1+\frac{2700}{50}\right)}{x}\\ =\frac{50\left(1+\frac{2700}{50.\infty }\right)}{1}\\ =\frac{50\left(1+0\right)}{1}\\ =50\end{array}$

So the limit of the = average cost function is $\underset{x\to \infty }{\lim }\overline{C}=\$50$

As the $\underset{x\to \infty }{\lim }\overline{C}=\$50$ for explaining the meaning of limit in context of the given problem,

This implies that as no. of tons of organic material becomes very large then the average cost per ton of recycling approaches to $\underset{x\to \infty }{\lim }\overline{C}=\$50$ .