Answer
(a) $360$ $ft/s$
(b) $0.096$ $rad/s$
Work Step by Step
(a)
by the pythagorean theorem
$4000^{2}+y^{2}$ = $l^{2}$
$2y\frac{dy}{dt}$ = $2l\frac{dl}{dt}$
$\frac{dy}{dt}$ = $600$ $ft/s$, $y$ = $3000$ $ft$
$l$ = $\sqrt {4000^{2}+3000^{2}}$ = $5000$ $ft$
$\frac{dl}{dt}$ = $\frac{y}{l}\frac{dy}{dt}$
$\frac{dl}{dt}$ = $\frac{3000}{5000}(600)$ = $360$ $ft/s$
(b)
$tanθ$ = $\frac{y}{4000}$
$sec^{2}θ\frac{dθ}{dt}$ = $\frac{1}{4000}\frac{dy}{dt}$
$\frac{dθ}{dt}$ = $\frac{cos^{2}θ}{4000}\frac{dy}{dt}$
$y$ = $3000$ $ft$, $\frac{dy}{dt}$ = $600$ $ft/s$, $l$ = $5000$ and $cosθ$ = $\frac{4000}{l}$ = $\frac{4000}{5000}$ = $\frac{4}{5}$
$\frac{dθ}{dt}$ = $\frac{(\frac{4}{5})^{2}}{4000}(600)$ = $0.096$ $rad/s$
