Answer
(b) $A=x^2$
(c) $ \frac{dA}{dt}=2x \frac{dx}{dt}$
(d)$\frac{dA}{dt}=12 ft^2/min$
Work Step by Step
(b) $ A=x\times x=x^2$
(c) $ \frac{dA}{dx} =\frac{d(x^2)} {dx}=2x $ ............ eq(1)
$\frac{dA}{dx}=\frac{dA}{dt}\frac{dt}{dx}$ ................. eq (2)
Putting equation (1) in equation (2)
$2x=\frac{dA}{dt}\frac{dt}{dx}$ Or
$ \frac{dA}{dt}\frac{dt}{dx}=2x$ Or
$\frac{dA}{dt}= \frac{2x} {\frac{dt}{dx}} =2x \frac{dx}{dt}$
(d) $ \frac{dA}{dt}=2x \frac{dx}{dt}$ .................. eq (3)
Putting $x=3$ and $\frac{dx}{dt}=2$ in equation (3)
$ \frac{dA}{dt}=2 \times 3 \times 2= 12 \frac{ft^2} {minute}$
