Answer
(a) $14cm^2/s$ (increasing)
(b) $0$ cm/sec (no change)
(c) $-\frac{14}{13}$ cm/sec (decreasing)
Work Step by Step
Identify the given quantities: $\frac{dl}{dt}=-2$ cm/sec, $\frac{dw}{dt}=2$ cm/sec, $l=12cm, w=5cm$
(a) Area: $A=lw$, $\frac{dA}{dt}=l\frac{dw}{dt}+w\frac{dl}{dt}=12(2)+5(-2)=14cm^2/s$ (increasing)
(b) Perimeter: $P=2(w+l)$, $\frac{dP}{dt}=2\frac{dw}{dt}+2\frac{dl}{dt}=2(2)+2(-2)=0$ cm/sec (no change)
(c) Length of diagonal: $D^2=l^2+w^2$, $2D\frac{dD}{dt}=2l\frac{dl}{dt}+2w\frac{dw}{dt}=2(12)(-2)+2(5)(2)=-28$
At $l=12, w=5$, we have $D=\sqrt {12^2+5^2}=13$, thus $\frac{dD}{dt}=-\frac{28}{2(13)})=-\frac{14}{13}$ (decreasing)
