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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