Question

A stone is dropped into a quiet lake and waves move in a circle at a speed of 3.5 cm/sec. At the instant when the radius of the circular wave is $7.5cm$. How fast is the enclosed area increasing?

• A. $32.5\pi c{m}^2/sec$
• B. $31.5\pi c{m}^2/sec$
• C. $52.5\pi c{m}^2/sec$
• D. None of these

Answer 1

Answer: (C)

The waves move in a circle at a speed of
$v=3.5cm/sec=\frac{dr}{dt}$
where $r$ is the instantaneous radius of circle.Let A be the area of circular wave at time $t$ Then $A=\pi r^2$
On differentiating both sides w.r.t. 't', we get
${\left[\frac{dA}{dt}\right]}_{{}_{{}_r}}=2\pi r\frac{dr}{dt}$
${\left[\frac{dA}{dt}\right]}_{{}_{{}_{7.5}}}=2\pi \times 7.5\times 3.5=52.5\pi$
$\therefore$ Rate of increasing area $52.5\pi c{m}^2/sec$