A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
The volume of a sphere (V) with radius (r) is given by,
\begin{align} V=\frac{4}{3}\pi r^2\end{align}
∴Rate of change of volume (V) with respect to time (t) is given by,
\begin{align} \frac{dV}{dt} =\frac{dV}{dr}.\frac{dr}{dt}\;\;\;[By\; Chain\; Rule]\end{align}
\begin{align} =\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right).\frac{dr}{dt}\end{align}
\begin{align} =4\pi r^2.\frac{dr}{dt}\end{align}
It is given that
\begin{align} \frac{dV}{dt}=900\; cm^3/s\end{align}
\begin{align} \therefore 900=4\pi r^2.\frac{dr}{dt}\end{align}
\begin{align} \Rightarrow \frac{dr}{dt}=\frac{900}{4\pi r^2}=\frac{225}{\pi r^2}\end{align}
Therefore, when radius = 15 cm,
\begin{align} \frac{dr}{dt}=\frac{225}{\pi (15)^2}=\frac{1}{\pi }\end{align}
Hence, the rate at which the radius of the balloon increases when the radius is 15 cm is
\begin{align} \frac{1}{\pi }\end{align}
NCERT questions are designed to test your understanding of the concepts and theories discussed in the chapter. Here are some tips to help you answer NCERT questions effectively:
Welcome to the NCERT Solutions for Class 12 Mathematics - Chapter . This page offers a step-by-step solution to the specific question from Excercise 1 , Question 8: A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic ce....
Comments