Question

In Fig. 10, a right circular cone of diameter r cm and height 12 cm rests on the base of a right circular cylinder of radius r cm. Their bases are in the same plane and the cylinder is filled with water upto a height of 12 cm. If the cone is then removed, find the height to which water level will fall.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{cone},R=\frac{\mathrm{r}}{2}, \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{cone},H=12\mathrm{cm}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{cylinder}=r\mathit{}\mathrm{and} \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{water}\mathrm{level}\mathrm{before}\mathrm{the}\mathrm{cone}\mathrm{is}\mathrm{taken}\mathrm{out},H=12\mathrm{cm} \\ \mathrm{Let}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{water}\mathrm{level}\mathrm{after}\mathrm{the}\mathrm{cone}\mathrm{is}\mathrm{taken}\mathrm{out}\mathrm{be}h. \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{water}\mathrm{in}\mathrm{the}\mathrm{cylinder}\mathrm{after}\mathrm{the}\mathrm{cone}\mathrm{is}\mathrm{taken}\mathrm{out}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}\mathrm{upto}\mathrm{a}\mathrm{height}\mathrm{of}12\mathrm{cm}-\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone} \\ \Rightarrow \mathrm{\pi }{r}^{2}h=\mathrm{\pi }{r}^{2}H-\frac{1}{3}\mathrm{\pi }{R}^{2}H \\ \Rightarrow \mathrm{\pi }{r}^{2}h=\mathrm{\pi }{r}^2\times 12-\frac{1}{3}\mathrm{\pi }{\left(\frac{r}{2}\right)}^2\times 12 \\ \Rightarrow \mathrm{\pi }{r}^{2}h=12\mathrm{\pi }{r}^2-\frac{1}{3}\mathrm{\pi }\times \frac{r^2}{4}\times 12 \\ \Rightarrow \mathrm{\pi }{r}^{2}h=12\mathrm{\pi }{r}^2-\mathrm{\pi }{r}^2 \\ \Rightarrow \mathrm{\pi }{r}^{2}h=11\mathrm{\pi }{r}^2 \\ \Rightarrow h=\frac{11\mathrm{\pi }{r}^2}{\mathrm{\pi }{r}^2} \\ \Rightarrow h=11\mathrm{cm} \\ \therefore \mathrm{The}\mathrm{fall}\mathrm{in}\mathrm{water}\mathrm{level}=12-11=1\mathrm{cm} \end{gathered}$$

So, the height to which the water level will fall is 1 cm.