Question

A solid right circular cone of height 60 cm and radius 30 cm is dropped in a right circular cylinder full of water, of height 180 cm and radius 60 cm. Find the volume of water left in the cylinder, in cubic metres. [CBSE 2015]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{height}\mathrm{of}\mathrm{cone},h=60\mathrm{cm}, \\ \mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{cone},r=30\mathrm{cm}, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{cylinder},H=180\mathrm{cm}\mathrm{and} \\ \mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylinder},R=60\mathrm{cm} \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{water}\mathrm{left}\mathrm{in}\mathrm{the}\mathrm{cylinder}=\mathrm{Volume}\mathrm{of}\mathrm{cylinder}-\mathrm{Volume}\mathrm{of}\mathrm{cone} \\ =\mathrm{\pi }{R}^{2}H-\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ =\frac{22}{7}\times 60\times 60\times 180-\frac{1}{3}\times \frac{22}{7}\times 30\times 30\times 60 \\ =\frac{22}{7}\times 30\times 30\times 60\left(2\times 2\times 3-\frac{1}{3}\right) \\ =\frac{22}{7}\times 54000\left(12-\frac{1}{3}\right) \\ =\frac{22}{7}\times 54000\times \frac{35}{3} \\ =1980000{\mathrm{cm}}^3 \\ =\frac{1980000}{1000000}{\mathrm{m}}^3 \\ =1.98{\mathrm{m}}^3 \end{gathered}$$

So, the volume of water left in the cylinder is 1.98 m³.