A solid sphere and a solid hemisphere have an equal total surface… - Vidyadip

Question

A solid sphere and a solid hemisphere have an equal total surface area. Prove that the ratio of their volume is $3 \sqrt{3}: 4$

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Answer

Total surface area of a sphere $=4 \pi r_1^2$ sq.units
Total surface area of a hemisphere $=3 \pi r_2^2$ sq.units
Ratio of Total surface area $=4 \pi r_1^2: 3 \pi r_2^2$
$1=\frac{4 \pi r _1^2}{3 \pi r _2^2} \quad \ldots($ Same Surface Area)
$
1=\frac{4 r _1^2}{3 r _2^2}
$
$\therefore \frac{ r _1^2}{ r _2^2}=\frac{3}{4}$
$r_1^2: r_2^2=3: 4$
$r_1: r_2=\sqrt{3}: 2$

Ratio of their volume
$
\begin{aligned}
& =\frac{4}{3} \pi r _1^3: \frac{2}{3} \pi r _2^3 \\
& =2 r _1^3: r _2^3 \\
& =2 \times(\sqrt{3})^3: 2^3 \\
& =2 \times 3 \sqrt{3}: 8 \ldots(\div 2) \\
& =3 \sqrt{3}: 4
\end{aligned}
$
Ratio of their volumes $=3 \sqrt{3}: 4$
Hence it is proved.

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