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 $7.5\, cm$. How fast is the enclosed area increasing?

• A. $32.5\, \pi\, cm^2/sec$
• B. $31.5\, \pi\, cm^2/sec$
• C. $52.5\, \pi\, cm^2/sec$
• D. None of these

Answer 1

Answer: (C)

The waves move in a circle at a speed of
$v = 3.5 \,cm/sec=\frac{dr}{dt}$
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
$\left[\frac{dA}{dt}\right]_{_{_r}}=2\pi r \frac{dr}{dt}$
$\left[\frac{dA}{dt}\right]_{_{_{7.5}}}=2\pi\times7.5\times3.5=52.5\,\pi$
$\therefore $ Rate of increasing area $52.5\, \pi\, cm^2/sec$