Q5:

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:

1. A vessel in the shape of an inverted cone with height $h_c = 8$ cm and radius $r_c = 5$ cm.

2. The vessel is filled with water to the brim.

3. Lead shots are spherical in shape with radius $r_s = 0.5$ cm.

4. When lead shots are dropped, $\frac{1}{4}$ of the water in the cone flows out.

To Find:

The number of lead shots ($n$) dropped into the vessel.

Step 1: Calculate the volume of the conical vessel.

The formula for the volume of a cone is $V_c = \frac{1}{3}\pi r_c^2 h_c$.

$V_c = \frac{1}{3} \times \pi \times (5)^2 \times 8$

$V_c = \frac{1}{3} \times \pi \times 25 \times 8$

$V_c = \frac{200}{3}\pi \text{ cm}^3$

Step 2: Determine the volume of water that flows out.

According to the problem, the volume of water that flows out is equal to $\frac{1}{4}$ of the total volume of the cone.

$V_{out} = \frac{1}{4} \times V_c$

$V_{out} = \frac{1}{4} \times \frac{200}{3}\pi = \frac{50}{3}\pi \text{ cm}^3$

Step 3: Calculate the volume of one spherical lead shot.

The formula for the volume of a sphere is $V_s = \frac{4}{3}\pi r_s^3$.

$V_s = \frac{4}{3} \times \pi \times (0.5)^3$

$V_s = \frac{4}{3} \times \pi \times 0.125$

$V_s = \frac{4}{3} \times \pi \times \frac{1}{8} = \frac{1}{6}\pi \text{ cm}^3$

Step 4: Formulate the equation to find the number of lead shots ($n$).

The volume of water displaced is equal to the total volume of the $n$ lead shots dropped into the vessel [By Archimedes' Principle].

$n \times V_s = V_{out}$

$n \times (\frac{1}{6}\pi) = \frac{50}{3}\pi$

Step 5: Solve for $n$.

Divide both sides by $\pi$:

$n \times \frac{1}{6} = \frac{50}{3}$

Multiply both sides by 6:

$n = \frac{50}{3} \times 6$

$n = 50 \times 2$

$n = 100$

Final Answer: The number of lead shots dropped in the vessel is 100.