A cylindrical vessel of height 5 cm and radius 4 cm is completely filled with sand, and the sand is then poured out to form a right circular cone of radius 6 cm. What will be the height of this cone, taking π = 22/7?

Introduction / Context:
This question examines your understanding of volume relationships between different solids, specifically a right circular cylinder and a right circular cone. In many practical situations, material is melted or transferred from one shape into another, and the volume must be conserved. Here, sand completely fills a cylinder and is then poured into a cone, and you must use volume conservation to determine the unknown height of the cone.

Given Data / Assumptions:

• Height of the cylindrical vessel, h_cyl = 5 cm.
• Radius of the cylinder, r_cyl = 4 cm.
• Sand is poured to form a right circular cone.
• Radius of the cone, r_cone = 6 cm.
• Height of the cone, h_cone, is unknown.
• Take π = 22/7 and assume there is no loss or gain of sand, so volumes are equal.

Concept / Approach:
We use the standard volume formulas for a cylinder and a cone. For a cylinder:
V_cylinder = π * r_cyl^2 * h_cyl. For a cone:
V_cone = (1 / 3) * π * r_cone^2 * h_cone. Because all the sand in the cylinder is transferred to the cone, the volumes must be equal. So we set V_cylinder = V_cone and solve for the unknown height h_cone.

Step-by-Step Solution:
Step 1: Write the cylinder volume: V_cylinder = π * r_cyl^2 * h_cyl. Step 2: Substitute r_cyl = 4 cm and h_cyl = 5 cm: V_cylinder = π * 4^2 * 5 = π * 16 * 5 = 80π. Step 3: Write the cone volume: V_cone = (1 / 3) * π * r_cone^2 * h_cone. Step 4: Substitute r_cone = 6 cm: V_cone = (1 / 3) * π * 6^2 * h_cone = (1 / 3) * π * 36 * h_cone = 12π * h_cone. Step 5: Set the volumes equal: 80π = 12π * h_cone. Step 6: Cancel π on both sides to simplify: 80 = 12 * h_cone. Step 7: Solve for h_cone: h_cone = 80 / 12 = 20 / 3 ≈ 6.67 cm. Step 8: Match this with the closest option, which is 6.67 cm.

Verification / Alternative check:
To verify, compute both volumes numerically. The cylinder volume is 80π. The cone volume using h_cone = 20 / 3 is V_cone = 12π * (20 / 3) = (240 / 3)π = 80π, which matches the cylinder volume exactly. This confirms that our calculation is correct and that no arithmetic mistake was made when solving for the height.

Why Other Options Are Wrong:
A height of 2.22 cm or 3.33 cm would produce a cone with much smaller volume than the original cylinder, so all the sand could not fit into the cone. A height of 1.67 cm is even smaller and clearly inconsistent with the large cylinder volume. The value 5 cm might be tempting because it matches the cylinder height, but volume relationships depend on both radius and height, and the cone has a different radius, so the heights cannot simply be equal.

Common Pitfalls:
One common error is to forget the factor of 1 / 3 in the cone volume formula. Another is to confuse the radii of the cylinder and the cone or to assume they are the same when they are not. Some students also try to compare heights directly rather than equating volumes. Finally, careless cancellation of π or incorrect simplification of fractions can lead to wrong numerical answers, especially under exam pressure.

Final Answer:
Therefore, the height of the cone formed by the sand is 6.67 cm.