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

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Detailed Solution

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, 2

r_{1}

=1.4cm
Radius

r_{1}

= 0.7cm
Cylindrical Beaker Diameter, 2

r_{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
⇒π

{r_{2}}^{2}

h = n

\times \frac{4 π {r_{1}}^{3}}{3}

Thus, n =

\frac{3 {r_{2}}^{2} h}{4 {r_{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
Hence, 150 marbles should be dropped into the beaker so that the water level rises by 5.6cm.

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