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

Question

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.

Accepted Answer

Step 1: Radius of cone $= 5$ cm, height of cone $= 8$ cm.
Volume of water in the cone $= \frac{1}{3} \pi 	imes (5)^2 	imes 8 = \frac{200\pi}{3}$ cm$^3$
Step 2: Volume of water flows out $= \frac{1}{4} 	imes \frac{200\pi}{3} = \frac{50\pi}{3}$ cm$^3$
Step 3: Radius of sphere (lead shot) $= 0.5 = \frac{1}{2}$ cm.
Volume of one lead shot $= \frac{4}{3} \pi 	imes \left(\frac{1}{2}ight)^3 = \frac{\pi}{6}$ cm$^3$
Step 4: Number of lead shots $= \frac{\frac{50\pi}{3}}{\frac{\pi}{6}} = \frac{50\pi}{3} 	imes \frac{6}{\pi} = 100$

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