Answer
$\$ 168.27$
Work Step by Step
see definition on p.281.
Average value of p =$\displaystyle \frac{1}{b-a}\int_{a}^{b}p(x)dx$
Using the given data,
Average value =$\displaystyle \frac{1}{50-40}\int_{40}^{50}\frac{90,000}{400+3x}dx$
$=\displaystyle \frac{90,000}{10}\int_{40}^{50}\frac{1}{400+3x}dx$
Find the indefinite integral first,
$\displaystyle \int\frac{1}{400+3x}dx=\left[\begin{array}{ll}
u=400+3x & \\
du=3dt & dt=\frac{1}{3}du
\end{array}\right]=\displaystyle \frac{1}{3}\int\frac{du}{u}$
$=\displaystyle \frac{1}{3}\ln|u|+C =\frac{1}{3}\ln|400+3x|+C$
Average value $=9,000\displaystyle \int_{40}^{50}\frac{1}{400+3x}dx=$
$=9,000\displaystyle \cdot\frac{1}{3}[\ln|400+3x|]_{40}^{50}$
$=3,000(\ln 550-\ln 520)\approx$168.268399953
