Question : A solid hemisphere has a radius 21 cm. It is melted to form a cylinder such that the ratio of its curved surface area to total surface area is 2 : 5, What is the radius (in cm) of its base (take $\pi=\frac{22}{7}$ )?
Option 1: 23
Option 2: 21
Option 3: 17
Option 4: 19
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Correct Answer: 21
Solution :
The radius of a solid hemisphere is 21 cm.
The ratio of the cylinder's curved surface area to its Total surface area is $\frac{2}{5}$.
The curved surface area of the cylinder = $2πRh$
The total surface area of cylinder = $2πR(R + h)$
The volume of the cylinder = $πR^2h$
The volume of the solid hemisphere = $\frac{2}{3}πr^3$
(where $r$ is the radius of a solid hemisphere and $R$ is the radius of a cylinder)
According to the question,
$\frac{\text{Curved Surface Area}}{\text{Total Surface Area}} = \frac{2}{5}$
⇒ $\frac{[2πRh]}{[2πR(R + h)]} = \frac{2}{5}$
⇒ $\frac{h}{(R + h)} = \frac{2}{5}$
⇒ $5h = 2R + 2h$
⇒ $h = (\frac{2}{3})R$
The cylinder's volume and the volume of a solid hemisphere are equal.
⇒ $πR^2h = (\frac{2}{3})πr^3$
⇒ $R^2 × (\frac{2}{3})R = (\frac{2}{3}) × (21)^3$
⇒ $R^3 = (21)^3$
⇒ $R$ = 21 cm
$\therefore$ The radius of its base is 21 cm
Hence, the correct answer is 21.
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