Question

Consider a rectangle whose length is increasing at the uniform rate of $2 m/sec$, breadth is decreasing at the uniform rate of $3 m/sec$ and the area is decreasing at the uniform rate of 5 $m^2$/sec. If after some time the breadth of the rectangle is $2 \,m$ then the length of the rectangle is

• A. $2\, m$
• B. $4 \,m$
• C. $1 \,m$
• D. $3\, m$

Answer 1

Answer: (D)

Let A be the area, b be the breadth and $\ell$ be the length of the rectangle.
Given : $\frac{dA}{dt} = -5,\, \frac{d\ell}{dt} = 2, \, \frac{db}{dt} = -3$
We know, $A = \ell \times b$
$\Rightarrow \frac{dA}{dt} = \ell. \frac{db}{dt} + b. \frac{d\ell}{dt} = 3\ell + 2b$
$\Rightarrow - 5 = —-3 \ell + 2b.$
When $b = 2$, we have
$-5 = -3\ell+4 \Rightarrow \ell = \frac{9}{3} = 3m$