The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2 cm/minute. When $x = 10$ cm and $y = 6$ cm, find the rate of change of the perimeter.

–2 cm/min

The correct answer is Option (3) → –2 cm/min

Since the length x of a rectangle is decreasing and the width y is increasing, we have

$\frac{dx}{dt}=-3\, cm/min$  ($\frac{dx}{dt}$ is -ve, because x is decreasing)

and $\frac{dy}{dt}= 2\, cm/min$ (given)

The perimeter, say P, of the rectangle at any time is given by

$P = 2(x + y)$, diff. w.r.t. t, we get

$\frac{dP}{dt}=2\left(\frac{dx}{dt}+\frac{dy}{dt}\right)=2(-3+2)\, cm/min = -2\, cm/min$.

When $x = 10 cm, y = 6 cm, \frac{dP}{dt}=-2\, cm/min$

Hence, the perimeter is decreasing at the rate of 2 cm/min when $x = 10$ cm and $y = 6$ cm.