Sushant has a vessel, of the form of an inverted cone, open at the top, of height $ 11 \mathrm{~cm} $ and radius of top as $ 2.5 \mathrm{~cm} $ and is full of water. Metallic spherical balls each of diameter $ 0.5 \mathrm{~cm} $ are put in the vessel due to which $ \left(\frac{2}{5}\right)^{\text {th }} $ of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?
Given:
Height of the conical vessel $=11\ cm$
Radius of the conical vessel $=2.5\ cm$
Diameter of each metallic spherical ball $=0.5\ cm$
\( \left(\frac{2}{5}\right)^{\text {th }} \) of the water in the vessel flows out.
To do:
We have to find the number of balls that were put in the vessel.
Solution:
Volume of water in the vessel $=\frac{1}{3} \pi \mathrm{R}^{2} h$
$=\frac{1}{3} \times \frac{22}{7} \times(2.5)^{2} \times 11$
$=\frac{22}{21} \times 6.25 \times 11 \mathrm{~cm}^{3}$
Volume of \( \frac{2}{5} \) th of the vessel $=\frac{22}{21} \times 6.25 \times 11 \times \frac{2}{5} \mathrm{~cm}^{3}$
Diameter of the spherical ball $=0.5 \mathrm{~cm}$
This implies,
Radius of the spherical ball $r=\frac{0.5}{2}$
$=0.25$
$=\frac{1}{4} \mathrm{~cm}$
Volume of each spherical ball $=\frac{4}{3} \pi r^{3}$
$=\frac{4}{3} \times \frac{22}{7} \times (\frac{1}{4})^{3}$
$=\frac{11}{168} \mathrm{~cm}^{3}$
Therefore,
Number of balls put in the vessel $=$ Volume of \( \frac{2}{5} \) th of the vessel $\div$ Volume of each spherical ball
$=\frac{\frac{2}{5} \times \frac{22}{21} \times 6.25 \times 11}{\frac{11}{168}}$
$=\frac{2 \times 22 \times 625 \times 11 \times 168}{5 \times 21 \times 100 \times 11}$
$=440$
$440$ balls were put in the vessel.
Sushant made the arrangement so that the water that flows out irrigates the flower beds. This shows the wise usage of water by Sushant.
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