A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of the base of the cone is 21 cm and its volume is 2/3 of the volume of the hemisphere, calculate the height of the c

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Solution

Q.
A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of the base of the cone is 21 cm and its volume is 2/3 of the volume of the hemisphere, calculate the height of the cone and the surface area of the toy.

Ans.
(5082)

We have, Radius of cone = Radius of hemisphere = 21 cm ⇒ r = 21 cm

According to question, Volume of cone $= \frac{2}{3} \times$ volume of hemisphere

$\Rightarrow \frac{1}{3} π r^{2} h = \frac{2}{3} \times \frac{2}{3} π r^{3} \Rightarrow h = \frac{4}{3} r = \frac{4}{3} \times 21 = 28 c m$

Slant height, $l = \sqrt{r^{2} + h^{2}} = \sqrt{21^{2} + 28^{2}} = 35 c m$

Total surface area $= C S A_{c o n e} + C S A_{h e m i s p h e r e} = π r l + 2 π r^{2} = π r (l + 2 r)$

$= \frac{22}{7} \times 21 \times (42 + 35) = 22 \times 3 \times 77 = 5082 c m^{2}$

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