Question

On dropping a stone in stationary water circular ripples are observed. The rate of flow of ripples is$6cm{s}^{-1}$. When the radius of the circle is$10cm$, then the fluid rate of increase in its area is

• A. $120c{m}^2{s}^{-1}$
• B. $12c{m}^2{s}^{-1}$
• C. $\pi c{m}^2{s}^{-1}$
• D. $120\pi c{m}^2{s}^{-1}$

Answer 1

The correct option is D

$120\pi c{m}^2{s}^{-1}$

Step 1. Given data:

1. The rate of flow of ripples$\frac{dr}{dt}$$=6cm{s}^{-1}$.
2. The radius of the circle$r=10cm$.

Step 2. Formula used:

The fluid rate of increase in area is given by,

$\frac{dA}{dt}=\frac{d\left(\pi r^2\right)}{dt}$ (where $A=\pi r^2$ the area of the water circular ripple)

Step 3. Calculating the fluid rate of increase in area

The rate of change of area with respect to time is,

$\begin{array}{rcl}\frac{dA}{dt}& =& \frac{d\left(\pi r^2\right)}{dt}\\ & =& 2\pi r\frac{dr}{dt}\\ & =& 2\pi \times 10\times 6c{m}^2{s}^{-1}\\ & =& 120\pi c{m}^2{s}^{-1}\end{array}$

Thus, the fluid rate of increase in area is$\begin{array}{rcl}& & 120\pi c{m}^2{s}^{-1}\end{array}$.

Hence, option D is the correct answer.