Question

The weekly marginal cost of producing x pairs of tennis shoes is given by the following function, where C(x) is cost in dollars.

C'(x)= 16+ (200/(x-1))

If the fixed costs are $2,000 per week, find the cost function. C(x)=??

What is the average cost per pair of shoes if 1,000 pairs of shoes are produced each week?

Answer 1

Answer:

The cost function is $C(x)=16x+200\ln \left|x-1\right|+2000$.

The average cost of 1000 shoes is $19.38.

Step-by-step explanation:

We define the marginal cost function to be the derivative of the cost function or $C'\left( x \right)$.

To find the cost function, $C(x)$ we need to integrate the marginal cost function

$\int {C'(x)} \, dx =C(x)\\\\\int \:16+\frac{200}{x-1}dx\\\\\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\=\int \:16dx+\int \frac{200}{x-1}dx$

$\int \:16dx=16x$

$\int \frac{200}{x-1}dx=200\ln \left|x-1\right|$

$\mathrm{Add\:a\:constant\:to\:the\:solution}\\\\C(x)=\int \:16+\frac{200}{x-1}dx=16x+200\ln \left|x-1\right|+D$

We know that the fixed costs are $2,000 per week, so the constant $D$ is equal to 2000, and the cost function is

$C(x)=16x+200\ln \left|x-1\right|+2000$

If $C(x)$ is the cost function for some item then the average cost function is,

                                                   $\overline{C}\left( x \right) = \frac{{C\left( x \right)}}{x}$

We know that 1,000 pairs of shoes are produced each week, so the the average cost is  

$\overline{C}\left( 1000 \right) = \frac{{C\left( 1000 \right)}}{1000}=\frac{16\cdot 1000+200\ln \left|1000-1\right|+2000}{1000} \\\\\overline{C}\left( 1000 \right)=\frac{200\ln \left(999\right)+18000}{1000}\approx19.38$