Question 9

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.

Answer

The volume of a sphere (*V*) with radius (*r*) is given by

\begin{align} V=\frac{4}{3 }\pi r^3\end{align}

Rate of change of volume (*V)* with respect to its radius (*r)* is given by,

\begin{align} \frac{dV}{dr }=\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right)=\frac{4}{3}\pi \left(3r^2\right)=4\pi r^2\end{align}

Therefore, when radius = 10 cm,

\begin{align} \frac{dV}{dr }=4\pi(10)^2=400\pi\end{align}

Hence, the volume of the balloon is increasing at the rate of 400π cm^{3}/s.

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