Answer
(a) $-12ft/s$ (down)
(b) $-\frac{119}{2}ft^2/s$ (decreasing)
(c) $-1$ rad/sec (decreasing).
Work Step by Step
Identify the given quantities: $\frac{dx}{dt}=5$ ft/sec, $x=12ft, l=13ft$
(a) $l^2=x^2+y^2$, $13^2=12^2+y^2$, $y=5ft$. Differentiate the first equation: with $\frac{dl}{dt}=0$, we have $2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$ or $12(5)+5\frac{dy}{dt}=0$; thus $\frac{dy}{dt}=-12ft/s$ (down)
(b) Area: $A=\frac{1}{2}xy$, $\frac{dA}{dt}=\frac{1}{2}(x\frac{dy}{dt}+y\frac{dx}{dt})=\frac{1}{2}(12(-12)+5(5))=-\frac{119}{2}ft^2/s$ (decreasing)
(c) $sin\theta=\frac{y}{13}$, $cos\theta\frac{d\theta}{dt}=\frac{y'}{13}=-\frac{12}{13}$. With $cos\theta=\frac{12}{13}$, we have $\frac{d\theta}{dt}=-\frac{12}{13}/cos\theta=-\frac{12}{13}\frac{13}{12}=-1$ rad/sec (decreasing).
