Question

The marginal average cost of producing x digital sports watches is given by the function C'^overbar(x), where C^overbar(x) is the average cost in dollars. C'^overbar(x) = 1, 200/x^2, C^overbar(100) = 22 Find the average cost function and the cost function. What are the fixes costs? The average cost function is C^overbar(x) = ____. The cost function m C(x) = ____. The fixed costs are $____.

Answer 1

Answer:

The average cost function is $\bar C(x)=-\frac{1200}{x}+34$.

The cost function is $C(x)=-1200+34x$.

The fixed costs are -$1200.

Step-by-step explanation:

If $y = C(x)$ is the total cost of producing x items, then the average cost, or cost per unit, is $\bar C(x)=\frac{C(x)}{x}$ and the marginal average cost is $\bar C'(x)=\frac{d}{dx}\bar C(x)$.

We know that the marginal average cost is given by

$\bar C'(x)=\frac{1200}{x^2}$

• To find the average cost function, we find the indefinite integral of $\frac{1200}{x^2}$ and determine the arbitrary integration constant using $\bar C(100)=22$

$\bar C'(x)=\frac{1200}{x^2}\\\\\bar C(x)=\int {\frac{1200}{x^2}} \, dx$

$\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx\\\\1200\cdot \int \frac{1}{x^2}dx\\\\\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1\\\\1200\cdot \frac{x^{-2+1}}{-2+1}\\\\-\frac{1200}{x}\\\\\mathrm{Add\:a\:constant\:to\:the\:solution}\\\\-\frac{1200}{x}+C$

Because $\bar C(100)=22$ C is

$22=-\frac{1200}{100}+C\\C=34$

And the average cost function is

$\bar C(x)=-\frac{1200}{x}+34$

• To find the cost function we multiply by x the average cost function according with the above definition

$x\cdot \bar C(x)=C(x)\\\\C(x)=(-\frac{1200}{x}+34)\cdot x\\\\C(x)=-1200+34x$

Cost functions are express as

$C=(fixed \:costs)+(variable \:costs)\\C=a+bx$

According with this definition and the cost function that we have the fixed costs are -$1200.