Question

A cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid cone of base diameter 7 cm and height 6 cm is completely immersed in water. Find the volume of water
(i) displaced out of the cylinder
(ii) left in the cylinder. [CBSE 2009]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Internal}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel},R=\frac{10}{2}=5\mathrm{cm}, \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel},H=10.5\mathrm{cm}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{cone},r=\frac{7}{2}=3.5\mathrm{cm}\mathrm{and} \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{cone},h=6\mathrm{cm} \\ \left(\mathrm{i}\right) \\ \mathrm{Volume}\mathrm{of}\mathrm{water}\mathrm{displaced}\mathrm{out}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{cone} \\ =\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ =\frac{1}{3}\times \frac{22}{7}\times 3.5\times 3.5\times 6 \\ =77{\mathrm{cm}}^3 \\ \left(\mathrm{ii}\right)\mathrm{As}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel}=\mathrm{\pi }{R}^{2}H \\ =\frac{22}{7}\times 5\times 5\times 10.5 \\ =825{\mathrm{cm}}^3 \\ \mathrm{So},\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{water}\mathrm{left}\mathrm{in}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel}-\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{cone} \\ =825-77 \\ =748{\mathrm{cm}}^3 \end{gathered}$$