Chapter 2, Problem 94RE

Solution

i.

To find the average cost including overhead cost of producing one shinguard in terms of x.

The average cost including overhead cost of producing one shinguard in terms of x is $A(x)=\$\frac{4.32x+4000}{x}$ .

Given:

Cost of a pair of shinguards $=\$5.25$ .

Cost to produce each pair of shinguards $=\$4.32$ .

Weekly overhead cost $=\$4000$ .

Concept Used:

Average cost is calculated as the total cost divided by the number of units of a good produced.

Calculation:

Let the number of shinguards produce in a week be x

The total cost to produce x shinguards in a week is,

  $C(x)=\$4.32x+4000$

Thus, average cost is,

  $A(x)=\$\frac{4.32x+4000}{x}$

Conclusion:

The average cost including overhead cost of producing one shinguard in terms of x is $A(x)=\$\frac{4.32x+4000}{x}$ .

b.

To find the number of shinguards to produce in order to make a profit of $\$8000$ .

The number of shinguards to produce in order to make a profit of $\$8000$ is approximately $12.90$ .

Given:

Profit, $P(x)=\$8000$ .

Concept Used:

Profit function is calculated as the difference of the revenue function and the cost function.

  $P(x)=R(x)-C(x)$

Calculation:

  $C(x)=\$4.32x+4000$

Revenue function, $R(x)=5.25x$ .

  $P(x)=\$8000$

Now, substitute the known values in the profit formula as,

  $\begin{array}{c}8000=5.25x-(4.32x+4000)\\ 8000=5.25x-4.32x-4000\\ 8000+4000=5.25x-4.32x\\ 12000=0.93x\\ x=12.90\end{array}$

Conclusion:

Hence, the number of shinguards to produce in order to make a profit of $\$8000$ is approximately $12.90$ .