UPSC NDA Math Model Paper 8

Given, at any instant t, for a sphere, r denotes the radius, S denotes the surface area.
Surface area S=4πr2
Differentiating w.r.to t, we get
⇒

ds

dt

=4π.2r

dr

dt

⇒

ds

dt

=8πr

dr

dt

⇒

dr

dt

=

1

8πr

ds

dt

....(1)
Volume of sphere V=

4

3

πr3
Differentiating w.r.to t, we get
⇒

dV

dt

=

4

3

π.3r2

dr

dt

⇒

dV

dt

=4πr2

dr

dt

From equation (1), we have
⇒

dV

dt

=4πr2

1

8πr

ds

dt

⇒

dV

dt

=

r

2

ds

dt

Hence, if at any instant t, for a sphere, r denotes the radius, S denotes the surface area and V denotes the volume, then

dV

dt

=

1

2

r

dS

dt