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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives3 Marks
Question
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10 cm$.
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Answer

Since, $V = \frac{4}{3}\pi {x^3}$
$\therefore \frac{{dV}}{{dx}} = \frac{d}{{dt}}\left( {\frac{4}{3}\pi {r^3}} \right)$
$= \frac{4}{3}\pi .3{r^2} = 4\pi {r^2}$
At x = 10 cm
$\Rightarrow \frac{{dV}}{{dx}} = 4\pi {\left( {10} \right)^2} = 400\pi $
Therefore, the volume is increasing at the rate of $400\pi\ cm^3/\sec$.

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