Question

A stone is dropped into a quiet lake and waves move in circles at the speed of $5$ cm/s At the instant when the radius of the circular wave is $8$ cm how fast is the enclosed area increasing?

• A. $84\pi c{m}^2/s$
• B. $80\pi c{m}^2/s$
• C. $90\pi c{m}^2/s$
• D. $96\pi c{m}^2/s$

Answer 1

Answer: (B)

$80\pi c{m}^2/s$

Given $\frac{dr}{dt}=5\text{cm/s}$
Area of wave is, $A=\pi r^2$
Thus rate of change of A is, $\frac{dA}{dt}=2\pi .r\frac{dr}{dt}$
Hence ${\left(\frac{dA}{dt}\right)}_{r=8cm}=2\pi \times 8\times 5=80\pi {cm}^2/s$