Q43 / 89 · KVPY 2017 (5 Nov SB/SX) · Mathematics
A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is
- Correct answerA$1: 1$
- B$1: 2$
- C$2: 1$
- D$\sqrt{2}: 1$
Solution
$$
\begin{array}{l}
\mathrm{S}=2 \pi \mathrm{R}^{2}+2 \pi \mathrm{Rh}+\pi \mathrm{R}^{2} \\
(\mathrm{R}=\text { radius of hemisphere \& cylinder) } \\
\mathrm{V}=\frac{2}{3} \pi \mathrm{R}^{3}+\pi \mathrm{R}^{2} \mathrm{~h} \\
\mathrm{~V}=\frac{2}{3} \pi \mathrm{R}^{3}+\pi \mathrm{R}^{2} \times\left(\frac{5-3 \pi \mathrm{R}^{2}}{2 \pi \mathrm{R}}\right) \\
\frac{\mathrm{dV}}{\mathrm{dR}}=2 \pi \mathrm{R}^{2}+\frac{5}{2}-\frac{9 \pi}{2} \mathrm{R}^{2}
\end{array}
$$
for maximum \& minimum $\frac{\mathrm{dV}}{\mathrm{dR}}=0$
$$
\begin{array}{l}
5 \pi R^{2}=S \\
5 \pi R^{2}=3 \pi R^{2}+2 \pi R h \\
2 R=2 h \\
h: R=1: 1
\end{array}
$$
Hence, the correct answer is option: 'A'.