Question

an accessories company finds that the revenue, in dollars, from the sale of x belts is given by $r(x) = 55x^{\frac{9}{10}}$. find the rate at which average revenue is changing when 174 belts have been produced and sold.
first, find the rate at which the average revenue is changing when x belts have been produced.
$\bar{r}(x) = \square$
when 174 belts have been produced and sold, the average revenue is changing at $\square$ for each additional belt.
(round to four decimal places as needed.)

Explanation:

Step1: Define average revenue

Average revenue $\bar{R}(x) = \frac{R(x)}{x} = \frac{55x^{9/10}}{x} = 55x^{-1/10}$

Step2: Differentiate average revenue

$\bar{R}'(x) = 55 \cdot (-\frac{1}{10})x^{-11/10} = -5.5x^{-11/10}$

Step3: Substitute x=174

$\bar{R}'(174) = -5.5 \cdot (174)^{-11/10} = -5.5 \cdot \frac{1}{174^{1.1}}$
Calculate $174^{1.1} \approx 174^{1 + 0.1} = 174 \cdot 174^{0.1} \approx 174 \cdot 1.713 \approx 298.062$
$\bar{R}'(174) \approx -5.5 / 298.062 \approx -0.01845$

Step4: Round to 4 decimals

$-0.0185$

Answer:

First blank: $-5.5x^{-11/10}$
Second blank: $-0.0185$