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, 1/4 of the water flows out. Find the number of lead shots dropped in the vessel.

To determine the number of lead shots dropped into the vessel, we first need to calculate the volume of the inverted cone and the volume of water that flows out when the lead shots are added.

Volume of the Inverted Cone

The formula for the volume of a cone is:

V = (1/3) × π × r² × h

Where:

• r = radius of the base (5 cm)
• h = height (8 cm)

Substituting the values:

V = (1/3) × π × (5)² × 8

V = (1/3) × π × 25 × 8 = (200/3)π cm³

Volume of Water That Flows Out

Since 1/4 of the water flows out, we calculate:

Volume of water out = (1/4) × (200/3)π = (50/3)π cm³

Volume of One Lead Shot

The volume of a sphere is given by:

V = (4/3) × π × r³

For each lead shot with a radius of 0.5 cm:

V = (4/3) × π × (0.5)³ = (4/3) × π × (1/8) = (1/6)π cm³

Calculating the Number of Lead Shots

Let n be the number of lead shots. The total volume of the lead shots is:

n × (1/6)π

Setting this equal to the volume of water that flows out:

n × (1/6)π = (50/3)π

Dividing both sides by π:

n × (1/6) = (50/3)

Multiplying both sides by 6:

n = 100

Final Answer

The number of lead shots dropped into the vessel is 100.