Applications Of Derivatives Question 14
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-
Options:
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} $
Show Answer
Answer:
Correct Answer: B
Solution:
[b] 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} $
$ \Rightarrow \frac{dV}{dt}=\frac{4}{3}\pi .3r^{2}\frac{dr}{dt}=4\pi r^{2}\frac{dr}{dt} $
$ =\frac{4\pi r^{2}}{8\pi r}.\frac{dS}{dt}=\frac{1}{2}r\frac{dS}{dt} $
BETA
