Answer
a. $=arccot \frac{x}{5}$
b. (x=10)$=\frac{2000}{125}$rad/hour
(x=3)$=\frac{2000}{34}$rad/hour
Work Step by Step
a.
\frac{d}{{dx}}\cot ^{ - 1} θ
$cot θ =\frac{adj}{opp} =\frac{x}{5}$
$θ=\cot ^{ - 1} (\frac{x}{5})$
$=arccot \frac{x}{5}$
b.
$\frac{dx}{dt}=-400$
$\frac{dθ}{dt}=\frac{-\frac{1}{5}}{1+\frac{x^2}{25}}\frac{dx}{dt}$
$=-\frac{5}{25+x^2}\frac{dx}{dt}$
when x=10,
$\frac{dθ}{dt}=-\frac{5}{25+100}(-400)$
$=\frac{2000}{125}$rad/hour
when x=3,
$\frac{dθ}{dt}=-\frac{5}{25+9}(-400)$
$=\frac{2000}{34}$rad/hour