Question

The length $x$ of a rectangle is decreasing at the rate of $5$ cm/minute and the width $y$ is increasing at the rate of $4$ cm/minute. When $x=8$ cm and $y=6$ cm, find the rates of change of the area of rectangle(in $c{m}^2$/min).

Answer 1

It is given that length $\left(x\right)$ is decreasing at the rate of $5$ cm/min

and the width $\left(y\right)$ is increasing at the rate of $4$ cm/min,

$\Rightarrow \frac{dx}{dt}=-5$ cm/min and $\frac{dy}{dt}=4$ cm/min

Thus the area $\left(A\right)$ of a rectangle is, $A=xy$

$\therefore {\left(\frac{dA}{dt}\right)}_{x=8,y=6}={(y\frac{dx}{dt}+x\frac{dy}{dt})}_{x=8,y=6}=6(-5)+8\left(4\right)=2$ cm${}^2$/min

Hence, the area is increasing at the rate of $2$ cm${}^2$/min.