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 $4cm/min$. If $x=8cm$ and $y=6cm$, then which of the following is correct?
I. The rate of change of the perimeter is $-2cm/min$.
II. The rate of change of the area of the rectangle is $12c{m}^2/min$.

• A. Only I is correct
• B. Only II is correct
• C. Both I and II are correct
• D. Both I and II are incorrect

Answer 1

Answer: (A)

At any instant time $t$, let length, breadth, perimeter and area of the rectangle are $x,y,P$ and A respectively,
then $P=2(x+y)$ and $A=xy$
It is given that $\frac{dx}{dt}=-5cm/min$ and
$\frac{dy}{dt}=4cm/min$
(- ve sign shows that the length is decreasing)
I. Now $P=2(x+y)$.
On differentiating w.r.t t, we get
$\frac{dP}{dt}=2\left(\frac{dx}{dt}+\frac{dy}{dt}\right)$
$=2\{-5+4\}cm/min=-2cm/min$
Hence, perimeter of the rectangle is decreasing at the rate of $2cm/min$.
II. Here, area of rectangle $A=xy$
On differentiating w.r.t. t, we get
Rate of change of area
$\frac{dA}{dt}=x\frac{dy}{dt}+y\frac{dx}{dt}$
$=8\times 4+6\times (-5)$
$=32-30=2c{m}^2/min$
Hence, area of the rectangle is increasing at the rate of $2c{m}^2/min$