A solid right circular cylinder of radius 7 cm and height 9 cm is melted and recast into a right circular cone of the same radius. What will be the height of this cone (in centimetres)?

Introduction / Context:
Here we apply volume conservation between a cylinder and a cone. When a solid is melted and recast into another shape, its volume remains the same. This question tests knowledge of volume formulas for a right circular cylinder and a right circular cone, and your ability to equate the two volumes when the radius is unchanged.

Given Data / Assumptions:

• Original solid: right circular cylinder.
• Cylinder radius r = 7 cm, cylinder height h₁ = 9 cm.
• New solid: right circular cone with the same radius r = 7 cm.
• Let the height of the cone be h₂ cm.
• Volume is conserved during melting and recasting.

Concept / Approach:
The volume of a cylinder is V_cyl = π * r^2 * h. The volume of a cone is V_cone = (1/3) * π * r^2 * h. Because the radius is the same in both shapes and the material is conserved, we set V_cyl equal to V_cone and solve for the unknown height of the cone. The common factors π and r^2 cancel out, leaving a simple linear equation in h₂.

Step-by-Step Solution:
Step 1: Compute the cylinder volume symbolically: V_cyl = π * r^2 * h₁ = π * 7^2 * 9 = π * 49 * 9 = 441π cubic centimetres. Step 2: Let the cone height be h₂. Volume of the cone: V_cone = (1/3) * π * r^2 * h₂ = (1/3) * π * 49 * h₂. Step 3: Set the volumes equal: 441π = (1/3) * π * 49 * h₂. Step 4: Cancel π from both sides to get 441 = (1/3) * 49 * h₂. Step 5: Multiply both sides by 3: 1323 = 49 * h₂. Step 6: Solve for h₂: h₂ = 1323 / 49. Step 7: Since 49 * 27 = 1323, we get h₂ = 27 cm.

Verification / Alternative check:
You can verify by inserting the found height into the cone volume formula. With h₂ = 27 cm, V_cone = (1/3) * π * 49 * 27 = (49 * 9) * π = 441π cubic centimetres, exactly equal to the cylinder volume calculated earlier. This confirms that no computational mistakes were made and that the height is correct.

Why Other Options Are Wrong:
If the height were 9 cm, the cone volume would be one third of the cylinder volume with the same radius, not equal to it. Heights 13.5 cm and 54 cm would give volumes that are either too small or too large compared to the cylinder volume. Only 27 cm produces a cone volume that exactly matches the melted cylinder volume.

Common Pitfalls:
A common error is forgetting the factor 1/3 in the cone volume formula, which can lead to setting π * r^2 * h₁ equal to π * r^2 * h₂ and concluding that h₂ = h₁. Another mistake is not cancelling π and r^2 properly, making the equation look more complicated than it really is. Keeping formulas clear and cancelling common factors early simplifies the algebra considerably.

Final Answer:
The height of the cone will be 27 cm.