Question 7

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

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:

\begin{align} \frac{dx}{dt} = -5 \;cm/min\; and \; \frac{dy}{dt}= 4 \;cm/min\end{align}

(a) The perimeter (*P)* of a rectangle is given by,

*P* = 2(*x + y*)

\begin{align} \therefore\frac{dp}{dt} = 2\left(\frac{dx}{dt} + \frac{dy}{dt}\right)= 2(-5+4)=-2\;cm/min\end{align}

Hence, the perimeter is decreasing at the rate of 2 cm/min.

(b) The area (*A)* of a rectangle is given by,

*A* = *x**⋅** y*

\begin{align} \therefore\frac{dA}{dt} = \frac{dx}{dt}.y + x.\frac{dy}{dt}=-5y + 4x \end{align}

When *x* = 8 cm and *y* = 6 cm,

\begin{align} \frac{dA}{dt} = (-5 \times 6 + 4 \times 8)\; cm^2/min = 2\; cm^2/min\end{align}

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

-->