Question

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

• A. 8
• B. 16
• C. 24
• D. 32

Answer 1

Answer: (A)

We have,
$\frac{dx}{dt}=-6,\frac{dy}{dt}=4$
Area of rectangle, $A=xy$
$\Rightarrow \frac{dA}{dt}=x\frac{dy}{dt}+y\cdot \frac{dx}{dt}$
$\Rightarrow \frac{dA}{dt}=x(4)+(-6)y$
$\Rightarrow \frac{dA}{dt}=4x-6y$
$\because$ Rate of change of area $(A)$ of rectangle i.e. $\frac{dA}{dt}$
$\therefore {\left(\frac{dA}{dt}\right)}_{(8,4)}=4(8)-6(4)=8$