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# A vessel is in the form of an inverted cone. Its height is 8cm and the radius of its top, which is open is 5cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5cm are dropped into the vessel, $\dfrac{1}{4}$ of the water flows out. Find the number of lead shots dropped in the vessel.

Last updated date: 19th Mar 2023
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Hint: - Volume of cone $= \dfrac{1}{3}\pi {\left( r \right)^2}h$
Given:
Height$\left( h \right)$of conical vessel$= 8cm$
Radius$\left( r \right)$ of conical vessel$= 5cm$
Radius$\left( {{r_1}} \right)$of the lead shots$= 0.5cm$
Let$x$number of lead shots were dropped in the vessel
Water spilled$= \dfrac{1}{4}$times of the volume of cone$= x \times$volume of spherical balls
As we know volume of cone is$= \dfrac{1}{3}\pi {\left( r \right)^2}h$
And volume of spherical balls$= \dfrac{4}{3}\pi r_1^3$
$\Rightarrow \dfrac{1}{4} \times \dfrac{1}{3} \times \pi {\left( r \right)^2}h = x \times \dfrac{4}{3}\pi r_1^3 \\ \Rightarrow \dfrac{1}{4} \times \dfrac{1}{3} \times \dfrac{{22}}{7}{\left( 5 \right)^2} \times 8 = x \times \dfrac{4}{3} \times \dfrac{{22}}{7}{\left( {0.5} \right)^3} \\ \Rightarrow \dfrac{{200}}{4} = 4x \times .125 \\ \Rightarrow x = \dfrac{{200}}{{16 \times .125}} = \dfrac{{200}}{2} = 100 \\$
So, the number of lead shots dropped into the vessel is equal to 100.