Chapter 11.4, Problem 57E

a)

To determine

To evaluate: the model for the average cost per unit produced.

Answer to Problem 57E

Model for the average cost per unit produced is given by $\frac{\text{Cost function}}{\text{Number of binoculars produced}}$ .

Explanation of Solution

Given:

A company’s cost function for producing a model of digital binoculars is given as $C=34x+18000$ where C is the cost (in dollars) and x is the number of binoculars produced.

Cost function is given as,

  $C=34x+18000$

Average cost is given by,

  $\begin{array}{c}\text{Average cost = }\frac{\text{Cost function}}{\text{Number of binoculars produced}}\\ =\frac{34x+18,000}{x}\end{array}$

So,

Model for the average cost per unit produced is given by $\frac{\text{Cost function}}{\text{Number of binoculars produced}}$ .

b)

To determine

To evaluate: the average costs per unit when x = 1000 and x =5000.

Answer to Problem 57E

The average costs per unit when x = 1000 is 52 dollars and the average costs per unit when x = 5000 is 188 dollars .

Explanation of Solution

Given:

A company’s cost function for producing a model of digital binoculars is given as $C=34x+18000$ where C is the cost (in dollars) and x is the number of binoculars produced.

Cost function is given as,

  $C=34x+18000$

Average cost is given by,

  $\begin{array}{c}\text{Average cost = }\frac{\text{Cost function}}{\text{Number of binoculars produced}}\\ =\frac{34x+18,000}{x}\end{array}$

So,

Average costs per unit when x = 1000,

  $\begin{array}{c}\text{Average cost = }\frac{34x+18000}{x}\\ =\frac{34\left(1000\right)+18000}{1000}\\ =52\text{ dollars}\end{array}$

Average costs per unit when x = 5000,

  $\begin{array}{c}\text{Average cost = }\frac{34x+18000}{x}\\ =\frac{34\left(5000\right)+18000}{1000}\\ =188\text{ dollars}\end{array}$

So,

The average costs per unit when x = 1000 is 52 dollars and the average costs per unit when x = 5000 is 188 dollars .

c)

To determine

To evaluate: the limit of the average cost function as x approaches infinity.

Answer to Problem 57E

The limit of average cost function is 34.

Explanation of Solution

Given:

A company’s cost function for producing a model of digital binoculars is given as $C=34x+18000$ where C is the cost (in dollars) and x is the number of binoculars produced.

Cost function is given as,

  $C=34x+18000$

Average cost is given by,

  $\begin{array}{c}\text{Average cost = }\frac{\text{Cost function}}{\text{Number of binoculars produced}}\\ =\frac{34x+18,000}{x}\end{array}$

So,

Finding the limit,

  $\begin{array}{c}\underset{x\to \infty }{\lim }\frac{34x+18000}{x}=\underset{x\to \infty }{\lim }34+\frac{18000}{x}\\ =34\end{array}$

The meaning of the limit in this context is when production of digital binoculars is infinity, then the cost per unit becomes constant that is 34.

So, the limit of average cost function is 34.