Question

Find rate of change of a circle
a) with respect to the radius r=10
b) with respect to the time when the radius is increasing at the rate of 0.7cm/s. Give that r=5

Answer 1

The circumference of a circle (C) with radius (r) is given by

C = 2πr.

Therefore, the rate of change of circumference (C) with respect to time (t) is given by,

dC/dt=dC/dr.dr/dt[ByChainRule]

=d/dr(2πr).d/rdt

 

=2π.dr/dt

It is given that

dr/dt=0.7cm/s

Hence, the rate of increase of the circumference 2π(0.7)=1.4π cm/s

 

The area of a circle (A) with radius (r) is given by,

A = πr2

Now, the rate of change of area (A) with respect to time (t) is given by,

dA/dt=d/dt(πr2).dr/dt=2πrdr/dt[ByChainRule]

It is given that,

dr/dt=0.7cm/s

∴dA/dt=2πr(.7)=1.4πr

Thus, when r = 10 cm,

dAdt=1.4π(10)=14πcm2/s

Hence, the rate at which the area of the circle is increasing when the radius is 10 cm is 14π cm2/s.