Choose the correct answer of the following question:

A solid is hemispherical at the bottom and conical (of same radius) above it. If the surface areas of the two parts are equal, then the ratio of its radius and the slant height of the conical part is

(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 4 : 1 [CBSE 2011]

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Solution

$Let the radius of the hemisphere or the radius of the cone be r and the slant height of the cone be l . Now, Surface area of the hemisphere = Surface area of the cone \Rightarrow 2 π r^{2} = π r l \Rightarrow \frac{π r^{2}}{π r l} = \frac{1}{2} \Rightarrow \frac{r}{l} = \frac{1}{2} ∴ r : l = 1 : 2$

Hence, the correct answer is option (a).

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A solid is hemispherical at the bottom and conical above it. If the surface areas of the two parts are equal, then the ratio of its radius and the height of its conical part is
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Let the radius of the hemisphere or the radius of the cone be r and the slant height of the cone be l.Now,Surface area of the hemisphere=Surface area of the cone⇒2πr2=πrl⇒πr2πrl=12⇒rl=12∴ r:l=1:2 Hence, the correct answer is option (a).