Question

A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into toys in the shape of a right circular cone mounted on a hemisphere of radius 3 cm. If the height of the toy is 12 cm, find the number of toys so formed.

Answer 1

Radius of cylinder = $\frac{12}{2}=6\mathrm{cm}$
Height of cylinder = 15 cm
Volume of cylinder$=\mathrm{\pi }{r}^{2}h$
$$\begin{gathered} =\left(\frac{22}{7}\times 6\times 6\times 15\right){\text{cm}}^3\left[d=12\text{cm},r\Rightarrow \frac{12}{2}\text{cm}=6\text{cm}\right] \\ =\frac{11880}{7}{\text{cm}}^3 \end{gathered}$$

Radius of the hemisphere = 3 cm

Height of the cone = Total height of the toy - Radius of the hemisphere

$=\left(12-3\right)\text{cm}=9\text{cm}$

Volume of each toy = (Volume of the hemisphere) + (Volume of the cone)
$$\begin{gathered} =\frac{2}{3}\mathrm{\pi }{r}^3+\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ =\frac{1}{3}\mathrm{\pi }{r}^2\left(2r+h\right) \\ =\frac{1}{3}\times \frac{22}{7}\times {\left(3\right)}^2\times \left[\left(2\times 3\right)+9\right]{\text{cm}}^3 \\ =\left(\frac{1}{3}\times \frac{22}{7}\times 9\times 15\right){\text{cm}}^3 \\ =\frac{990}{7}{\text{cm}}^3 \end{gathered}$$

Number of toys$=\frac{\text{Volume of the cylinder}}{\text{Volume of each toy}}$

$$\begin{gathered} =\left(\frac{11880\times 7}{7\times 990}\right) \\ =12 \end{gathered}$$