question 7 8 pts solve the problem. an appliance manufacturer has determined that the cost, in dollars, of…

question 7 8 pts solve the problem. an appliance manufacturer has determined that the cost, in dollars, of producing x espresso makers is given by c(x)=3800 + 1.2x^0.3. if the revenue from the sale of x espresso makers is given by r(x)=71x^0.5 find the rate at which the average profit per espresso maker is changing when 50 espresso makers have been made and sold. round to the nearest cent. edit view insert format tools table 12pt paragraph b i u a t2

question 7 8 pts solve the problem. an appliance manufacturer has determined that the cost, in dollars, of producing x espresso makers is given by c(x)=3800 + 1.2x^0.3. if the revenue from the sale of x espresso makers is given by r(x)=71x^0.5 find the rate at which the average profit per espresso maker is changing when 50 espresso makers have been made and sold. round to the nearest cent. edit view insert format tools table 12pt paragraph b i u a t2

Answer

Explanation:

Step1: Define profit function

Profit $P(x)=R(x)-C(x)=71x^{0.5}-(3800 + 1.2x^{0.3})=71x^{0.5}-3800 - 1.2x^{0.3}$

Step2: Define average - profit function

Average - profit $\overline{P}(x)=\frac{P(x)}{x}=\frac{71x^{0.5}-3800 - 1.2x^{0.3}}{x}=71x^{- 0.5}- \frac{3800}{x}-1.2x^{-0.7}$

Step3: Differentiate the average - profit function

Using the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $\overline{P}'(x)=71\times(-0.5)x^{-1.5}+3800x^{-2}-1.2\times(-0.7)x^{-1.7}$ $=-35.5x^{-1.5}+3800x^{-2}+0.84x^{-1.7}$

Step4: Evaluate at $x = 50$

$\overline{P}'(50)=-35.5\times50^{-1.5}+3800\times50^{-2}+0.84\times50^{-1.7}$ $=-35.5\times\frac{1}{50^{1.5}}+3800\times\frac{1}{2500}+0.84\times\frac{1}{50^{1.7}}$ $=-35.5\times\frac{1}{353.553}+1.52 + 0.84\times\frac{1}{1060.66}$ $=- 0.1004+1.52+0.00079$ $\approx1.42$

Answer:

$$1.42$