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
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
Updated On: Jul 28, 2023
- $\frac{9}{7}$
- $\frac{7}{9}$
- $\frac{2}{9}$
- $\frac{9}{2}$
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The Correct Option is C
Solution and Explanation
$V = \frac{4}{3}\pi r^{3}\quad\quad4500 \pi = \frac{4\pi r^{3}}{3}$
$\frac{dV}{dt} = 4\pi r^{2} \left(\frac{dr}{dt}\right)\quad\quad45 ??25 ??3 = r^{3}$
$r = 15\, m$
after $49$ min $= \left(4500 - 49.72\right)\pi = 972\, \pi\,m^{3}$
$972\,\pi = \frac{4}{3} \pi r^{3}$
$r^{3} = 3??43 = 3??3^{5}$
$r = 9$
$72\, \pi = 4\pi ??9 ??9 \left(\frac{dr}{dt}\right)$
$\frac{dr}{dt} = \left(\frac{2}{9}\right)$
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Application of Derivatives
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