Question

Tthe area of a circle increases at a rate of 5 cm^2/s.

Answer 1

Answer:

(a) 0.198 cm/s (b) 1 cm/s

Step-by-step explanation:

It is given that,

The rate of change of area of a cicle, $\dfrac{dA}{dt}=5\ cm^2/s$

(a) The area of a circle is given by :

$A=\pi r^2$

Differentiate both sides,

$\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\\\\\text{Put the value of dA/dt and r = 4 cm}\\\\\dfrac{dr}{dt}=\dfrac{dA/dt}{2\pi r}\\\\\dfrac{dr}{dt}=\dfrac{5}{2\pi \times 4}\\\\\dfrac{dr}{dt}=0.198\ cm/s$

(b) Circumference, $C=2\pi r$ = 5 cm

$\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\\\\\dfrac{dr}{dt}=\dfrac{dA/dt}{2\pi r}\\\\\dfrac{dr}{dt}=\dfrac{5}{5}\\\\\dfrac{dr}{dt}=1\ cm/s$

Hence,

(a) 0.198 cm/s (b) 1 cm/s