Question

A toy is in the shape of a hemisphere of same base radius. If the volume of the toy is $231c{m}^2$ and it's diameter is $7cm$. find the height of the toy.

Answer 1

Given: Volume of toy $=231c{m}^3$

Diameter of a hemisphere $=7cm$

radius of hemisphere $=\frac{\text{Dia}}{2}=\frac{7}{2}=3.5cm$

Cone and hemisphere have equal radius.

radius of hemisphere $=$ radius of cone $=3.5cm$

Height of hemisphere $=$ radius of hemisphere $=3.5cm$

Let $H$ be the height of toy.

$H=$ height of cone $+$ height of hemisphere

$H=h+r$ , ( $h=$ height of cone)

$H=h+3.5$

Volume of toy $=$ volume of cone $+$ volume of hemisphere

Volume of toy $=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3$

Volume of toy $=\frac{\pi r^2}{3}(h+2r)$

$\Rightarrow 231=\frac{22\times (3.5{)}^2}{7\times 3}(h+2\times 3.5)$

$\Rightarrow 231=\frac{22\times 7\times 7}{7\times 2\times 2\times 3}(h+7)$ $[\because 3.5=\frac{7}{2}]$

$\Rightarrow 231=\frac{11\times 7\times 1}{1\times 1\times 2\times 3}(h+7)$

$\Rightarrow 231=\frac{77}{6}(h+7)$

$\Rightarrow \frac{231\times 6}{77}=h+7$

$\Rightarrow 18=h+7$

$\Rightarrow 18-7=h$

$\Rightarrow h=11cm$
Height of toy $H=h+r$

Height of toy $=11+3.5=14.5$

Height of toy $=14.5cm$

Hence, the height of the toy $H=14.5cm$