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 $x=8cm$ and $y=6cm$ find the rates of change of (a) perimeter, and (b) the area of the rectangle.

Answer 1

(i)

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 perimeter $\left(P\right)$ of a rectangle is, $P=2(x+y)$
$\frac{dP}{dt}=2(\frac{dx}{dt}+\frac{dy}{dt})=2(-5+4)=-2$ cm/min
Hence, the perimeter is decreasing at the rate of $2$ cm/min.

(ii)

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.