Chapter 11.4, Problem 57E

(a)

To determine

To calculate:

Model for average cost per unit produced.

Answer to Problem 57E

Model for the average cost per unit produced is $\overline{C}=\frac{73x+25000}{x}$

Explanation of Solution

Given information:

  $C=73x+25000$

Where,

  $C=$ Cost in dollars.

Calculation:

Here, let’s assume $\overline{C}$ be the average cost per unit produced and $x$ be the amount of material then,

  $\overline{C}=\frac{C}{x}$

But the cost of function for cost per unit produced $C=73x+25000$

Then,

  $\begin{array}{l}\overline{C}=\frac{C}{x}\\ =\frac{73x+25000}{x}\end{array}$

The required model for the average cost per unit produced recycle is $\overline{C}=\frac{73x+25000}{x}$

(b)

To determine

To calculate:

Average costs per unit.

Answer to Problem 57E

Average cost of recycling $1000PDA\text{'}s$ and $5000PDA\text{'}s$ of organic material is $\$73250$ and $\$365005$ .

Explanation of Solution

Given information:

  $C=73x+25000$

Where,

  $C=$ Cost in dollars.

  $\begin{array}{l}x=1000\\ x=5000\end{array}$

Calculation:

For determining the average cost per unit,

Average cost $\overline{C}$ for $x$ is

  $\overline{C}=\frac{73x+25000}{x}$

So average cost for $1000PDA\text{'}s$ is:

  $\begin{array}{l}\overline{C}=\frac{73\left(1000\right)+25000}{\left(1000\right)}\\ =\frac{73000+25000}{100}\\ =\frac{98000}{100}\\ =73250\end{array}$

Thus, required average cost of recycling $1000PDA\text{'}s$ of organic material is $\$73250$

So average cost for $5000PDA\text{'}s$ is:

  $\begin{array}{l}\overline{C}=\frac{73\left(5000\right)+25000}{\left(5000\right)}\\ =\frac{7300+25000}{100}\\ =\frac{390000}{5000}\\ =365005\end{array}$

Thus, required average cost of recycling $100PDA\text{'}s$ of organic material is $\$365005$

(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 57E

Limit of the average cost function is $\underset{x\to \infty }{\lim }\overline{C}=\$73$

Explanation of Solution

Given information:

  $C=73x+25000$

Where,

  $C=$ Cost in dollars.

Calculation:

To find out the Limit of average cost function,

Average cost function $\overline{C}$ for $x$ approaches to infinity.

For the rational function $f\left(x\right)=\frac{N\left(x\right)}{D\left(x\right)}$

Where,

  $N\left(x\right)=a_n{x}^n+\mathrm{...}+a_0$

  $D\left(x\right)=b_m{x}^m+\mathrm{...}+b_0$

The limit $x$ approaches positive or negative infinity as as below:

  $\underset{x\to \infty }{\lim }f\left(x\right)=\{\begin{array}{l}0,ifn<m\\ \frac{a_n}{b_m},ifn=m\end{array}$

The limit does not exist if $n>m$

The degree of the numerator is equals to the degree of the denominator. O the limit of the function is the ratio of the coefficients of the highest powered terms.

  $\begin{array}{l}\underset{x\to \infty }{\lim }\overline{C}=\underset{x\to \infty }{\lim }\frac{73x+25000}{x}\\ =\frac{73}{1}\\ =73\end{array}$

The limit of function $\overline{C}$ as $x$ approaches to infinity is $73$ .

Thus, as the no. of PDA gets very large the average cost per unit produces is $\$73$