QJEEM12Mada18: JEE Main 2012 Mathematics - Class 12 JEEM12Mada18 Solution: A spherical balloon is filled with cubic meters of helium gas.…

Question

Verified Solution • 2025–26 Session moderate

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 :

Accepted Answer

Volume of spherical balloon

= V = {4 \over 3}\pi {r^3}

\Rightarrow 4500\pi = {{4\pi {r^3}} \over 3}

( as Given, volume

= 4500\pi {m^3}

)

Differentiating both the sides,

w.r.t't'

we get,

{{dV} \over {dt}} = 4\pi {r^2}\left( {{{dr} \over {dt}}} \right)

Now, it is given that

{{dV} \over {dt}} = 72\pi

\therefore

After

49

min, Volume -

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {4500 - 49 \times 72} \right)\pi

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {4500 - 3528} \right)\pi

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 972\pi {m^3}

\Rightarrow V = 972\,\,\pi {m^3}

\therefore

972\pi = {4 \over 3}\pi r{}^3

\Rightarrow {r^3} = 3 \times 243 = 3 \times {3^5} = {3^6} = {\left( {{3^2}} \right)^3} \Rightarrow r = 9

Also, we have

{{dV} \over {dt}} = 72\pi

\therefore

72\pi = 4\pi \times 9 \times 9\left( {{{dr} \over {dt}}} \right) \Rightarrow {{dr} \over {dt}} = \left( {{2 \over 9}} \right)

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