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BITSAT 2022
Exam
BITSAT
Year
2022
Subject
Mathematics
Chapter
Application of Derivatives

BITSAT 2022 · Mathematics · Application of Derivatives

A spherical balloon is filled with 4500 π cubic meters of helium...

A spherical balloon is filled with 4500π4500 \pi cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π72 \pi cubic meters per minute then the rate (in meters per minute) at which the radius of the balloon decreases 49min49 \mathrm{ min} after the leakage began is

Options

  1. A97\frac{9}{7}
  2. B79\frac{7}{9}
  3. C29\frac{2}{9}
  4. D99

Correct answer

C. 29\frac{2}{9}

Step-by-step solution

V=43πr3⇒4500π=4πr33⇒r=15mV=\frac{4}{3} \pi r^3 \Rightarrow 4500 \pi=\frac{4 \pi r^3}{3}\Rightarrow \quad r=15 \mathrm{ m} After 49min⁡=(4500−49.72)π=972πm349 \min =(4500-49.72) \pi=972 \pi \mathrm{m}^3⇒972π=43πr3⇒r3=3×243=3×35\begin{aligned} \Rightarrow & & 972 \pi & =\frac{4}{3} \pi r^3 \\ \Rightarrow & & r^3 & =3 \times 243=3 \times 3^5\end{aligned}$\begin{aligned} r & =9 \\ \frac{d v}{d t} & =4 \pi r^2\left(\frac{d r}{d t}\right) \\ 72 \pi & =4 \pi \times 9 \times 9\left(\frac{d r}{d t}\right) \\ \frac{d r}{d t} & =\left(\frac{2}{9}\right)\end{aligned}
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