Question : The total surface area of a solid metallic hemisphere is 462 cm2. This is melted and moulded into a right circular cone. If the radius of the base of the cone is the same as that of the hemisphere, then its height is: (use $\pi=\frac{22}{7}$)
Option 1: 14 cm
Option 2: 7 cm
Option 3: 21 cm
Option 4: 28 cm
Correct Answer: 14 cm
Solution :
The total surface area of a solid metallic hemisphere = 462 cm
2
Let the radius of the hemisphere be $r$ cm.
So, $3 \pi r^2$ = 462
⇒ $r$ = $\sqrt{\frac{462 × 7}{22 × 3}}$ = $\sqrt{49}$ = 7
So, the volume of the hemisphere = $\frac{2}{3} \pi r^3$ = $\frac{2}{3}×\frac{22}{7}×7^3$ = $\frac{2156}{3}$
Now, the radius of hemisphere ($r$) = radius of cone ($R$)
So, The volume of the cone = volume of the hemisphere
⇒ $\frac{2156}{3}$ = $\frac{1}{3} \pi R^2 H$ (Let $H$ be the height of the cone)
⇒ $\frac{2156}{3}$ = $\frac{1}{3}×\frac{22}{7}×7^2×H$
$\therefore H$ = 14 cm
Hence, the correct answer is 14 cm.
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