If at any instant t for a sphere r denotes the radius s denotes the surface area and v denotes the volume then what is `(d)/(dt)` equal to ?

Options

A. `1/2 S (dr)/(dt)`
B. `1/2 r (ds)/(dt)`
C. `r (ds)/(dt)`
D. `1/2 r^(2) (ds)/(dt)`

Answer

C

Explanation

Surface area of spere `S=4pir^(2)`
Differntialte both sides w.r.t ltrbgt `rarr (ds)/(dt)=(87pidr)/(dt) rarr (dr)/(dt) =(1)/(8pir).(ds)/(dt)`
`rarr (dv)/(dt)=4/3 pi . 3r^(2)(dr)/(dt)=4pi r^(2) (dr)/(dt)`
`=(4 pi r^(2))/(8 pi r). (ds)/(dt)=1/2 r (ds)/(dt)`