Question

Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm, containing some water. Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6 cm.

Answer 1

Diameter of each marble = 1.4 cm
Radius of each marble = 0.7 cm
Volume of each marble $=\frac{4}{3}{\mathrm{\pi r}}^3=\frac{4}{3}\mathrm{\pi }\times {\left(0.7\right)}^3{\mathrm{cm}}^3$

The water rises as a cylindrical column.
Volume of cylindrical column filled with water$={\mathrm{\pi r}}^2\mathrm{h}=\mathrm{\pi }\times {\left(\frac{7}{2}\right)}^2\times 5.6{\mathrm{cm}}^3$

Total number of marbles

$$\begin{gathered} =\frac{\mathrm{Volume}\mathrm{of}\mathrm{cylindrical}\mathrm{water}\mathrm{column}}{\mathrm{Volume}\mathrm{of}\mathrm{marble}} \\ =\frac{\mathrm{\pi }\times {\left(\frac{7}{2}\right)}^2\times 5.6}{\frac{4}{3}\pi \times {\left(0.7\right)}^3} \\ =\frac{7\times 7\times 5.6\times 3}{2\times 2\times 4\times 0.7\times 0.7\times 0.7} \\ =150 \end{gathered}$$