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

Given:

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

Diameter of a hemisphere= 7 cm

cone and hemisphere have equal radius.

radius of hemisphere = radius of cone = 3.5 cm

Height of hemisphere= radius of hemisphere= 3.5 cm

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 $=\frac{1}{3}\pi r^2(h+2r)$

$231=\left(\frac{22}{7}\right)\times (3.5{)}^2\times (\frac{1}{3}\left)\right(h+2\times 3.5)$

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

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

$h+7=\frac{(3\times 3)}{(.5\times .5\times 2)}$

$h+7=\frac{900}{50}=\frac{90}{5}=18$

h+7= 18

h=18-7

h= 11 cm

Height of toy = h+r

Height of toy = 11+3.5= 14.5

Height of toy =14.5 cm

Hence, the height of the toy = 13.5 cm