solve the problem. a mens suit manufacturer finds that the cost, in dollars, of producing x suits is given…

solve the problem. a mens suit manufacturer finds that the cost, in dollars, of producing x suits is given by c(x)=882 + 13x^0.5. find the rate at which the average cost is changing when 200 suits have been produced. round the answer to four decimal places. edit view insert format tools table 12pt paragraph b i u a t2

solve the problem. a mens suit manufacturer finds that the cost, in dollars, of producing x suits is given by c(x)=882 + 13x^0.5. find the rate at which the average cost is changing when 200 suits have been produced. round the answer to four decimal places. edit view insert format tools table 12pt paragraph b i u a t2

Answer

Explanation:

Step1: Define average - cost function

The average - cost function $\overline{C}(x)$ is given by $\overline{C}(x)=\frac{C(x)}{x}$, where $C(x)=882 + 13x^{0.5}$. So, $\overline{C}(x)=\frac{882}{x}+13x^{- 0.5}$.

Step2: Differentiate the average - cost function

Using the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $\overline{C}'(x)=- \frac{882}{x^{2}}-\frac{13}{2}x^{-1.5}$.

Step3: Evaluate the derivative at $x = 200$

Substitute $x = 200$ into $\overline{C}'(x)$. $\overline{C}'(200)=-\frac{882}{200^{2}}-\frac{13}{2}\times200^{-1.5}$ $=-\frac{882}{40000}-\frac{13}{2}\times\frac{1}{200^{1.5}}$ $=-\frac{882}{40000}-\frac{13}{2}\times\frac{1}{200\sqrt{200}}$ $=-\frac{882}{40000}-\frac{13}{2}\times\frac{1}{200\times10\sqrt{2}}$ $=-\frac{882}{40000}-\frac{13}{4000\sqrt{2}}$ $\approx - 0.0221 - 0.0023$ $=-0.0244$

Answer:

$-0.0244$