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 231 cm³ and its diameter is 7 cm, then find the height of the toy. [CBSE 2012]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Base}\mathrm{radius}\mathrm{of}\mathrm{cone}=\mathrm{Base}\mathrm{radius}\mathrm{of}\mathrm{hemisphere}=r=\frac{7}{2}=3.5\mathrm{cm}, \\ \mathrm{As},\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{toy}=231{\mathrm{cm}}^3 \\ \Rightarrow \mathrm{Volume}\mathrm{of}\mathrm{cone}+\mathrm{Volume}\mathrm{of}\mathrm{hemisphere}=231 \\ \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^{2}h+\frac{2}{3}\mathrm{\pi }{r}^3=231 \\ \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^2\left(h+2r\right)=231 \\ \Rightarrow \frac{1}{3}\times \frac{22}{7}\times 3.5\times 3.5\times \left(h+2\times 3.5\right)=231 \\ \Rightarrow \frac{38.5}{3}\times \left(h+7\right)=231 \\ \Rightarrow h+7=\frac{231\times 3}{38.5} \\ \Rightarrow h+7=18 \\ \Rightarrow h=18-7 \\ \Rightarrow h=11\mathrm{cm} \\ \mathrm{So},\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{toy}=h+r=11+3.5=14.5\mathrm{cm} \end{gathered}$$