Question

A spherical balloon is filled with $4500\pi $ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi $ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is:
$\begin{align}
  & \left( \text{a} \right)\text{ }\dfrac{6}{7} \\
 & \left( b \right)\text{ }\dfrac{4}{9} \\
 & \left( c \right)\text{ }\dfrac{2}{9} \\
 & \left( d \right)\text{ None of these} \\
\end{align}$

Answer 1

Hint: The volume of a spherical balloon is given by $V=\dfrac{4}{3}\pi {{r}^{3}}$, so find the volume after $49$ minutes. After that, put the volume after $49$ minutes in the above formula and you will get radius and then differentiate w.r.t to $t$ and substitute the values. You will get the answer.

Complete step by step answer:
The volume of spherical balloon is given by $V=\dfrac{4}{3}\pi {{r}^{3}}$ . ………(1)
Initial volume given is $=4500\pi $cubic meters.
After 49 minutes the volume will be,
$=4500\pi -49\times 72\pi =972\pi $ .
Equating this with the volume of the balloon from equation (i), we get
$V=\dfrac{4}{3}\pi {{r}^{3}}=972\pi $
On cross-multiplying and solving, we get
r = 9metre
This is the radius of the balloon after 49 minutes of leakage.
Now differentiating equation (1) w.r.t to ‘t’, we get,
$\dfrac{dv}{dt}=\dfrac{4\pi }{3}\times 3{{r}^{2}}\dfrac{dr}{dt}$
Substituting the given values, we get
$\begin{align}
  & 72\pi =\dfrac{4\pi }{3}\times 3{{(9)}^{2}}\dfrac{dr}{dt} \\
 & 72=4(81)\dfrac{dr}{dt} \\
 & \dfrac{72}{4\times 81}=\dfrac{dr}{dt} \\
\end{align}$
$\dfrac{dr}{dt}=\dfrac{2}{9}$
So we get the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage begins is $\dfrac{2}{9}$meters per minute.
The correct answer is option (C).

Note: Read the question and see what is asked. Your concept regarding mensuration should be clear. A proper assumption should be made. Do not make silly mistakes while substituting. Equate it in a proper manner do not confuse yourself.
Student often make mistake when calculating the decrease in volume. The subtract initial volume with $72\pi $.
But this is wrong, the volume is decreasing at $72\pi $ cubic meters per minute, so after 49 minutes the volume decreased will be $72\pi \times 49$.