Question

The length of a rectangle is decreasing at the rate of $3\text{cm}/\text{min}$ and its width is increasing at the rate of $2\text{cm}/\text{min}.$ If length is $10\text{cm},$ width is $6\text{cm}$ and $P,A$ represent the perimeter and area of the rectangle respectively, then which of the following is/are true

• A. $\frac{dP}{dt}=-2{\text{cm}}^2/\text{min}$
• B. $\frac{dA}{dt}=2{\text{cm}}^2/\text{min}$
• C. $\frac{dP}{dt}=2\text{cm}/\text{min}$
• D. $\frac{dP}{dt}=-2\text{cm}/\text{min}$

Answer 1

Answer: (B), (D)

$\frac{dP}{dt}=-2\text{cm}/\text{min}$

$\text{Given,}\frac{dx}{dt}=-3\text{cm}/\text{min},\frac{dy}{dt}=2\text{cm}/\text{min}x=10\text{cm},y=6\text{cm}$
$\text{(i)}P=2(x+y)\Rightarrow \frac{dP}{dt}=2(\frac{dx}{dt}+\frac{dy}{dt})=-2\text{cm}/\text{min}$
$\text{(ii)}A=xy\Rightarrow \frac{dA}{dt}=(x\frac{dy}{dt}+y\frac{dx}{dt})=10\left(2\right)+6(-3)=2{\text{cm}}^2/\text{min}$