Applications Of Derivatives Question 33
Question: If the rate of change in volume of spherical soap bubble is uniform, then the rate of change of surface area varies as
Options:
A) Square of radius
B) Square root of radius
C) Inversely proportional to radius
D) Cube of the radius
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Let volume $ =V=\frac{4}{3}\pi r^{3} $ -(1) and surface area $ =S=4\pi r^{2} $ -(2) Now, $ (1)\Rightarrow \frac{dv}{dt}=\frac{4}{3}\times 3\pi r^{2}\times \frac{dr}{dt} $
$ =4\pi r^{2}\frac{dr}{dt}…(3) $
$ (2)\Rightarrow \frac{ds}{dt}=4\pi \times 2\times r\frac{dr}{dt}=\frac{8\pi r^{2}}{r}\frac{dr}{dt} $
$ =\frac{2}{r}[ 4\pi r^{2}\frac{dr}{dt} ]=\frac{2}{r}\frac{dv}{dt} $ (from 3)
BETA
