An inverted conical vessel is completely filled with water up to its brim. The height of the cone is 8 centimetres and the radius of the open circular top is 5 centimetres. Solid spherical metallic balls, each of radius 1/2 centimetre, are dropped into the vessel and 25% of the water overflows. How many balls were dropped into the vessel?

Introduction / Context:
This is a volume displacement problem involving a cone filled with water and small spherical balls that cause overflow when dropped into the water. It tests your understanding of volume formulas for cones and spheres, as well as the physical principle that the volume of water displaced by submerged objects equals the volume of those objects. In this case, the displaced water appears as overflow.

Given Data / Assumptions:

• The vessel is an inverted right circular cone.
• Height of the cone h = 8 cm.
• Radius of the top of the cone R = 5 cm.
• The cone is initially filled completely with water up to the brim.
• Solid spherical metallic balls of radius r_ball = 1/2 cm are dropped into the water.
• After dropping some balls, 25% of the water overflows.
• No water remains above the brim; the cone is still full in volume, but part of the water has been replaced by balls.
• We must determine the number of balls dropped.

Concept / Approach:
The key idea is conservation of volume and displacement. The volume of water that overflows equals the volume of the balls that are submerged, assuming all are fully below the water surface when the system stabilises. Initially, the cone contains a volume V_cone of water. When balls are added, 25% of the initial water volume flows out. This overflow volume is equal to the total volume of the balls. Therefore:
Total volume of balls = 25% of V_cone = (1 / 4) * V_cone.
We compute V_cone using the cone volume formula and the volume of one ball using the sphere volume formula, then equate N * V_ball = (1 / 4) * V_cone to find N.

Step-by-Step Solution:
Step 1: Compute the volume of the cone. V_cone = (1 / 3) * pi * R^2 * h. Step 2: Substitute R = 5 cm and h = 8 cm: V_cone = (1 / 3) * pi * 5^2 * 8. Step 3: 5^2 = 25, so V_cone = (1 / 3) * pi * 25 * 8 = (200 / 3) * pi cubic centimetres. Step 4: Since 25% of the water overflows, overflow volume = (1 / 4) * V_cone = (1 / 4) * (200 / 3) * pi = (50 / 3) * pi cubic centimetres. Step 5: Compute the volume of one spherical ball. V_ball = (4 / 3) * pi * r_ball^3. Step 6: Radius of each ball r_ball = 1/2 cm, so r_ball^3 = (1/2)^3 = 1/8. Step 7: Thus V_ball = (4 / 3) * pi * (1 / 8) = (4 / 24) * pi = (1 / 6) * pi cubic centimetres. Step 8: Let N be the number of balls. Then total volume of balls = N * V_ball = N * (1 / 6) * pi. Step 9: Equate the total volume of balls to the overflow volume: N * (1 / 6) * pi = (50 / 3) * pi. Step 10: Cancel pi from both sides to get N / 6 = 50 / 3. Step 11: Multiply both sides by 6: N = 6 * (50 / 3) = 2 * 50 = 100. Step 12: Therefore, 100 balls were dropped into the vessel.

Verification / Alternative check:
We can check by computing the total volume of 100 balls. V_total_balls = 100 * (1 / 6) * pi = (100 / 6) * pi = (50 / 3) * pi cubic centimetres. This matches exactly the 25% overflow volume computed for the cone, confirming that the calculation of N is correct and consistent with volume conservation.

Why Other Options Are Wrong:
200 and 400: These would give total ball volumes that are double or four times the required overflow volume, resulting in more water displaced than the specified 25%.
150 and 250: These also lead to total volumes inconsistent with exactly one quarter of the cone's volume, either exceeding or falling short of the required displacement.

Common Pitfalls:
Some students mistakenly take 25% of the cone's height or radius instead of its volume, which is incorrect. Others forget that the volume of a sphere depends on the cube of the radius and may incorrectly use (1/2)^2 instead of (1/2)^3. Confusing diameter and radius when computing ball volume is another common error. Carefully applying the correct volume formulas and remembering that overflow volume equals displaced volume helps avoid these mistakes.

Final Answer:
The number of balls dropped into the vessel is 100.