Vessel Shape and Dimensions:
The vessel is in the form of an inverted cone with a height of 18 meters. The top radius of the vessel is 50 cm, and it tapers down to a point.

Filling the Vessel:
The vessel is filled with water up to a certain level. Let's assume this level is h meters from the top of the vessel.

Volume of the Vessel:
To find the number of lead shots in the vessel, we need to calculate the volume of water in the vessel. Since the vessel is in the shape of an inverted cone, we can use the formula for the volume of a cone.

The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14159), r is the radius of the cone's base, and h is the height of the cone.

In this case, the radius of the vessel at the top is 50 cm, which is equal to 0.5 meters. The height of the vessel is given as 18 meters.

Volume of Water:
To find the volume of water in the vessel, we need to find the height of the water level h. Given that the water fills up to the top of the vessel, which has a radius of 0.5 cm, we can use similar triangles to find the height of the water level.

The ratio of the height of the water level to the height of the vessel is equal to the ratio of the radius of the water level to the radius of the vessel.

Let's denote the radius of the water level as r1. Using the ratio mentioned above, we can write the equation: (18 - h) / 18 = r1 / 0.5.

Simplifying this equation, we get: r1 = 0.5 * (18 - h) / 18.

Calculating the Volume:
Now that we have the radius of the water level, we can substitute it into the formula for the volume of a cone.

The volume of water in the vessel is given by V = (1/3) * π * r1^2 * h.

Calculating the Number of Lead Shots:
To find the number of lead shots, we need to divide the volume of water by the volume of each lead shot.

Given that the radius of each lead shot is 0.5 cm, we can use the formula for the volume of a sphere to find the volume of each lead shot.

The formula for the volume of a sphere is V = (4/3) * π * r^3, where V is the volume and r is the radius of the sphere.

Substituting the radius of each lead shot into the formula, we get V_shot = (4/3) * π * (0.5)^3.

Finally, we can calculate the number of lead shots by dividing the volume of water by the volume of each lead shot:

Number of lead shots = V_water / V_shot.

Conclusion:
By following the above steps, you can calculate the number of lead shots in the vessel. Remember to convert the units consistently throughout the calculations to get