Search any question & find its solution
Question:
Answered & Verified by Expert
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 $\frac{d V}{d t}$ equal to?
Options:
what is $\frac{d V}{d t}$ equal to?
Solution:
1023 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{2} r \frac{d S}{d t}$
Surface area of sphere $\mathrm{S}=4 \pi r^{2}$ Differentiate both sides w.r.t. $\mathrm{t}^{\prime}$
$\Rightarrow \frac{\mathrm{dS}}{\mathrm{dt}}=\frac{8 \pi \mathrm{rdr}}{\mathrm{dt}} \Rightarrow \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}=\frac{1}{8 \pi \mathrm{r}} \cdot \frac{\mathrm{d} \mathrm{s}}{\mathrm{dt}}$
and Volume $=\mathrm{V}=\frac{4}{3} \pi \mathrm{r}^{3}$
$\Rightarrow \quad \frac{\mathrm{dV}}{\mathrm{dt}}=\frac{4}{3} \pi \cdot 3 \mathrm{r}^{2} \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}=4 \pi \mathrm{r}^{2} \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}$
$\quad=\frac{4 \pi \mathrm{r}^{2}}{8 \pi \mathrm{r}} \cdot \frac{\mathrm{d} \mathrm{S}}{\mathrm{dt}}=\frac{1}{2} \mathrm{r} \frac{\mathrm{dS}}{\mathrm{dt}}$
$\Rightarrow \frac{\mathrm{dS}}{\mathrm{dt}}=\frac{8 \pi \mathrm{rdr}}{\mathrm{dt}} \Rightarrow \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}=\frac{1}{8 \pi \mathrm{r}} \cdot \frac{\mathrm{d} \mathrm{s}}{\mathrm{dt}}$
and Volume $=\mathrm{V}=\frac{4}{3} \pi \mathrm{r}^{3}$
$\Rightarrow \quad \frac{\mathrm{dV}}{\mathrm{dt}}=\frac{4}{3} \pi \cdot 3 \mathrm{r}^{2} \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}=4 \pi \mathrm{r}^{2} \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}$
$\quad=\frac{4 \pi \mathrm{r}^{2}}{8 \pi \mathrm{r}} \cdot \frac{\mathrm{d} \mathrm{S}}{\mathrm{dt}}=\frac{1}{2} \mathrm{r} \frac{\mathrm{dS}}{\mathrm{dt}}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.