Q52 / 155 · WBJEE 2019 · Mathematics

If the radius of a spherical balloon increases by $0.1 \%$, then its volume increases approximately by

  • A
    $0.2 \%$
  • Correct answer
    B
    $0.3 \%$
  • C
    $0.4 \%$
  • D
    $0.05 \%$

Solution

Let $V$ be the volume of spherical ballon of radius $r$ Then, $V=\frac{4}{3} \pi r^{3}$ $\Rightarrow \quad \log V=\log \left(\frac{4 \pi}{3}\right)+3 \log r \Rightarrow \frac{1}{V} \frac{d V}{d r}=0+\frac{3}{r}$ $\Rightarrow \quad \frac{1}{V} \frac{d V}{d r}=\frac{3}{r} \Rightarrow \frac{1}{V} \frac{\Delta V}{\Delta r}=\frac{3}{r}$ $\Rightarrow \quad \frac{\Delta V}{V}=3 \frac{\Delta r}{r}$ $\Rightarrow \frac{\Delta V}{V} \times 100=3 \frac{\Delta r}{r} \times 100$ $\Rightarrow \frac{\Delta V}{V} \times 100=3 \times 01=0.3$ $\therefore$ Percentage increase in volume is $0.3 \%$

Hence, the correct answer is option: 'B'.

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