Question

A toy is in the shape of a cone mounted on a hemisphere of same base radius. If the volume of the toy is $231c{m}^3$ and its diameter is $7cm$, find the height of the toy.

• A.
  $7cm$
• B.
  $7.5cm$
• C.
  ​​​​​​​$11cm$
• D.
  ​​​​​​​$14.5cm$

Answer 1

The correct option is D

​​​​​​​$14.5cm$

Given:

Volume of the toy = 231 $c{m}^3$
Diameter of the hemisphere = 7 cm

The cone and hemisphere have equal radii.
$\Rightarrow$ Radius of hemisphere = Radius of cone = $r$ = 3.5 cm

Height of hemisphere = Radius of hemisphere = 3.5 cm
Let $H$ be the height of toy.
$\Rightarrow$ $H$ = height of cone + height of hemisphere
$\Rightarrow$ $H$ = $h$ + $r$, ($h$ = height of cone)
$\Rightarrow$ $H$ = $h$ + 3.5

Now, Volume of toy = Volume of cone + Volume of hemisphere
$\Rightarrow$ Volume of toy $=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3$
$\Rightarrow$ Volume of toy $=\frac{1}{3}\pi r^2(h+2r)$
$\Rightarrow 231=\left(\frac{1}{3}\right)\times \frac{22}{7}\times \left(3.5{)}^2\right(h+2\left(3.5\right))$
$\Rightarrow$ $h+7=\frac{(231\times 3\times 7)}{(3.5\times 3.5\times 22)}$
$\Rightarrow$ $h+7=18$
$\Rightarrow$ $h=11cm$

Thus, height of the toy = $h$ + $r$
= 11 $cm$ + 3.5 $cm$
= 14.5 $cm$