Question

The radius of a circular puddle is growing at a rate of 25 cm/s.

Answer 1

Answer:

a) 2500π cm/s

b) 300√π cm/s

Step-by-step explanation:

Given:

Rate of growth of radius, $\frac{dr}{dt}$ = 25 cm/s

Area of circle is given as:

A = πr²

a)Rate of growth of area, $\frac{dA}{dt}=\frac{d(\pi r^2)}{dt}$

or

⇒ $\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}$ ............(1)

at r = 50 cm

on substituting the respective values, we get

⇒ $\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}$

or

⇒ $\frac{dA}{dt}$ = 2π(50)25 = 2500π cm/s

b) when area , A = 36 cm²

36 = πr²

r = $\frac{6}{\sqrt{\pi}}$

thus, using (1)

⇒ $\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}$

on substituting the respective values, we get

⇒ $\frac{dA}{dt}=(2)\pi (\frac{6}{\sqrt{\pi}})25$

or

⇒ $\frac{dA}{dt}$ = 300√π cm/s