Question

A toy is in the form of a cone mounted on a hemisphere of diameter 7 cm. The total height of the toy is 14.5 cm. Find the volume and the total surface area of the toy.

Answer 1

A figure representing the toy is shown below:


Height of cone = 14.5 cm - 3.5 cm = 11 cm
The slant height of the conical part can be expressed using Pythagoras' theorem as shown:

$l=\sqrt{{\left(\frac{7}{2}\right)}^2+{\left(11\right)}^2}=11.54cm$

Volume of the toy = Volume of the conical part + Volume of the hemisphere
Therefore,
$\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{toy}=\frac{2}{3}{\mathrm{\pi r}}^3+\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=\frac{1}{3}{\mathrm{\pi r}}^2\left(2\mathrm{r}+\mathrm{h}\right)=\frac{1}{3}\mathrm{\pi }{\left(\frac{7}{2}\right)}^2\left(2\times \frac{7}{2}+11\right)=231{\mathrm{cm}}^3$

Total area of the toy = Curved surface area of the cone + Curved surface area of the hemisphere.
Therefore,

$\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{toy}=2{\mathrm{\pi r}}^2+\mathrm{\pi rl}=\mathrm{\pi r}\left(2\mathrm{r}+\mathrm{l}\right)=\mathrm{\pi }\times \left(\frac{7}{2}\right)\times \left(7+11.54\right)=203.94{\mathrm{cm}}^2$