Answer
a. $F(x)=500\sec^{2}x$
b. $\approx 827$ N
Work Step by Step
a.
F and $\sec^{2}x$ are proportional: $F(x)=k\sec^{2}x$
$(\sec 0=1)$
$F(0)=k\cdot 1=500\quad\Rightarrow\quad k=500$
$F(x)=500\sec^{2}x$
b.
Average value $=\displaystyle \frac{1}{b-a}\int_{a}^{b}F(x)dx$
$\displaystyle \frac{1}{\pi/3-0}\int_{0}^{\pi/3}500\sec^{2}xdx$
$=\displaystyle \frac{1500}{\pi}[\tan x]_{0}^{\pi/3}$
$=\displaystyle \frac{1500}{\pi}(\sqrt{3}-0)$
$\approx 827$ N
