Question

A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm² .The cone is placed
upon the cylinder. Volume of the solid figure so formed is 500 cm³ . Find the total height of the figure.

Answer 1

Height of the cylinder, h = 3 cm

Let the radius of the cylinder be r cm and the height of the cone be H cm.

Area of the base of cylinder = 100 cm²

$\therefore \mathrm{\pi }{r}^2=100$ .....(1)

Volume of the solid figure = 500 cm³

∴ Volume of the cylinder + Volume of the cone = 500 cm³

$$\begin{gathered} \Rightarrow \mathrm{\pi }{r}^{2}h+\frac{1}{3}\mathrm{\pi }{r}^{2}H=500 \\ \Rightarrow \mathrm{\pi }{r}^2\left(h+\frac{H}{3}\right)=500 \\ \Rightarrow 100\left(3+\frac{H}{3}\right)=500\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow 3+\frac{H}{3}=\frac{500}{100}=5 \\ \Rightarrow \frac{H}{3}=5-3=2 \\ \Rightarrow H=6\mathrm{cm} \end{gathered}$$
∴ Total height of the figure = h + H = 3 + 6 = 9 cm

Thus, the total height of the figure is 9 cm.