A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
Given:
A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out.
To do:
We have to find the number of lead shots dropped in the vessel.
Solution:
Height of the cone $=8 \mathrm{~cm}$
The radius of the cone $=5 \mathrm{~cm}$
This implies,
Volume of the cone $=\frac{1}{3} \pi r^{2} h$
$=\frac{1}{3} \pi(5)^{2} \times 8$
$=\frac{200}{3} \pi \mathrm{cm}^{3}$
The radius of one spherical lead shot $=0.5 \mathrm{~cm}$
Therefore,
Volume of one spherical lead shot $=\frac{4}{3} \pi r^{3}$
$=\frac{4}{3} \pi(0.5)^{3}$
$=\frac{4 \times 0.125}{3} \pi \mathrm{cm}^{3}$
$=\frac{0.5}{3} \pi \mathrm{cm}^{3}$
When spherical lead is dropped in the vessel, one-fourth of the water flows out
Let the number of lead shots be $n$.
Volume of $n$ spherical shots $=\frac{1}{4}$ volume of conical vessel
$n(\frac{0.5}{3} \pi)=\frac{1}{4}(\frac{200}{3} \pi)$
$n(0.5)=50$
$n=\frac{50 \times 10}{5}$
$n=100$
The number of lead shots dropped in the vessel is 100.
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