Question
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 $$\frac{{dV}}{{dt}}$$ equal to ?
A.
$$\frac{1}{2}S\frac{{dr}}{{dt}}$$
B.
$$\frac{1}{2}r\frac{{dS}}{{dt}}$$
C.
$$r\frac{{dS}}{{dt}}$$
D.
$$\frac{1}{2}{r^2}\frac{{dS}}{{dt}}$$
Answer :
$$\frac{1}{2}r\frac{{dS}}{{dt}}$$
Solution :
Surface area of sphere $$S = 4\pi {r^2}$$
Differentiate both sides w.r.t. $$t'$$
$$ \Rightarrow \frac{{dS}}{{dt}} = \frac{{8\pi rdr}}{{dt}}$$
and Volume $$ = V = \frac{4}{3}\pi {r^3}$$
$$\eqalign{
& \Rightarrow \frac{{dV}}{{dt}} = \frac{4}{3}\pi .3{r^2}\frac{{dr}}{{dt}} \cr
& = 4\pi {r^2}\frac{{dr}}{{dt}} \cr
& = \frac{{4\pi {r^2}}}{{8\pi r}}.\frac{{dS}}{{dt}} \cr
& = \frac{1}{2}r\frac{{dS}}{{dt}} \cr} $$