Question

The length $x$, of a rectangle is decreasing at the rate of $5cm/min$ and the width $y$, is increasing at the rate of $4cm/min$. When $x=8cm$ and $y=6cm$, find the rate of change of area of the rectangle.

Answer 1

Let $A$ be the area of rectangle and $x$ and $y$ be the length and width of the rectangle respectively.

Then $A=xy$

$\therefore \frac{dA}{dt}=x\frac{dy}{dt}+y\frac{dx}{dt}$

$\left[\begin{array}{cc}\because x=8cm,& \frac{dx}{dt}=-5and\frac{dy}{dt}=4andy=6cm\end{array}\right]$

$\therefore$ Rate of change of area $=32-30$

= $2c{m}^2/min$