# Class 12 Mathematics Application of Derivatives: NCERT Solutions for Question 9

This page focuses on the detailed Application of Derivatives question answers for Class 12 Mathematics Application of Derivatives, addressing the 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.'. The solution provides a thorough breakdown of the question, highlighting key concepts and approaches to arrive at the correct answer. This easy-to-understand explanation will help students develop better problem-solving skills, reinforcing their understanding of the chapter and aiding in exam preparation.
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.

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 cm3/s.