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.
100 lead shots
Explanation
This is a volume displacement problem. When lead shots are dropped into the water-filled cone, they displace water equal to their own volume. Since one-fourth of the water flows out, the total volume of lead shots equals one-fourth of the cone's volume. By dividing this displaced volume by the volume of one lead shot (sphere), we get the number of shots.
Solution Steps
Step 1: Calculate volume of water in the conical vessel Volume of cone = (1/3)πr²h = (1/3) × π × 5² × 8 = (1/3) × π × = (200π/3) cm³
Step 2: Calculate volume of water that flows out Since one-fourth of water flows out, Volume of water displaced = (1/4) × (200π/3) = (50π/3) cm³
Step 3: Calculate volume of one lead shot (sphere) Volume of sphere = (4/3)πr³ = (4/3) × π × (0.5)³ = (4/3) × π × 0.125 = (π/6) cm³
Step 4: Find the number of lead shots Number of lead shots = (Volume of water displaced) ÷ (Volume of one lead shot) = (50π/3) ÷ (π/6) = (50π/3) × (6/π) = = 100 lead shots