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 $(a)$ the perimeter, and $(b)$ the area of the rectangle.

Answer 1

The correct answer is $2 cm^2/min.$
Since the length $(x)$ is decreasing at the rate of $5 cm/minute$ and the width $(y)$ is increasing at the rate of $4 cm/minute$, we have:
$\frac{dx}{dt}=-5cm/min$ and $\frac{dy}{dt}=4 cm/min$
(a) The perimeter $(P)$ of a rectangle is given by, $P = 2(x + y)$
$∴ \frac{dp}{dt}=2(\frac{dx}{dt}+\frac{dy}{dt})=2(-5+4)=-2 cm/min.$
(b) The area $(A)$ of a rectangle is given by, $A = x \times y$
$∴\frac{dA}{dt}=\frac{dx}{dt}.y+x.\frac{dy}{dt}=-5y+4x$
When $x = 8 cm$ and $y = 6 cm, \frac{dA}{dt}=(-5\times6+4\times8)cm^2/min=2cm^2/min$
Hence, the area of the rectangle is increasing at the rate of $2 cm^2/min.$