Question

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

Answer 1

Let $m$ be the number of marbles needed, Let $r_1$ and $r_2$ be the radii of the marbles and cylindrical beaker respectively.

Marbles:
Diameter, $2r_1=1.4cm$
Radius $r_1=0.7cm$

Cylindrical Beaker:

Diameter, $2r_2=7cm$

Radius, $r_2=\frac{7}{2}cm$

Let $h$ be the height of the water level raised. Then, $h=5.6cm$

After the marbles are dropped into the beaker,

Volume of water raised $=$ Volume of $n$ marbles

$\Rightarrow \pi r_2^{2}h=n\times \frac{4}{3}\pi r_1^3$

Thus, $n=\frac{3r_2^{2}h}{4r_1^3}$

$n=\frac{3\times \frac{7}{2}\times \frac{7}{2}\times 5.6}{4\times \frac{7}{10}\times \frac{7}{10}\times \frac{7}{10}}=150$

The number of marbles needed is $150$.