A balloon which always remains spherical, has a variable diameter… - Vidyadip

Question

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}\left( {2x + 1} \right)$ Find the rate of change of its volume with respect to x.

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Answer

Given: Diameter of the balloon $= \frac{3}{2}\left( {2x + 1} \right)$

$\therefore$ Radius of the balloon $= \frac{3}{4}\left( {2x + 1} \right)$

$\therefore$ Volume of the balloon $= \frac{4}{3}\pi {\left( {\frac{3}{4}\left( {2x + 1} \right)} \right)^3}$

$= \frac{{9\pi }}{{16}}{\left( {2x + 1} \right)^3}$ cube units

$\therefore$ Rate of change of volume w.r.t. $x = \frac{{dV}}{{dx}}$

$= \frac{{9\pi }}{{16}}.3{\left( {2x + 1} \right)^2}.\frac{d}{{dx}}\left( {2x + 1} \right)$

$= \frac{{27\pi }}{16}{\left( {2x + 1} \right)^2}.2$

$= \frac{{27\pi }}{8}{\left( {2x + 1} \right)^2}$

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