There is a cylinder circumscribing the hemisphere such that their bases are common and their height is same. The respective ratio (hemisphere : cylinder) of volume and curved surface area are:

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Solution

Q.
There is a cylinder circumscribing the hemisphere such that their bases are common and their height is same. The respective ratio (hemisphere : cylinder) of volume and curved surface area are:

(i) Ratio of volume = 1 : 3 (ii) Ratio of CSA = 1 : 2

(iii) Ratio of volume = 2 : 3 (iv) Ratio of CSA = 1 : 1

Choose the correct option from the following:

1 (i) and (iv)

2 (ii) and (iii)

3 (i) and (iii)

4 (iii) and (iv)

Ans.
(4)

Height of cylinder = r
Radius of sphere = r

$V o l u m e o f c y l i n d e r = V_{1} = π (r^{2}) (r) = π r^{3}$

$V o l u m e o f h e m i s p h e r e = V_{2} = \frac{2}{3} π r^{3}$

$∴ \frac{V_{1}}{V_{2}} = \frac{π r^{3}}{\frac{2}{3} π r^{3}} = \frac{3}{2} ∴ V_{2} : V_{1} = 2 : 3$

$R a t i o o f C S A = \frac{CSA of hemisphere}{CSA of cylinder} = \frac{2 π r^{2}}{2 π r (r)} = 1 : 1$

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