A solid sphere of radius 9 cm is melted and recast to form a smaller solid sphere of radius 6 cm and a right circular cylinder with the same radius 6 cm. What is the height of the cylinder (in centimetres)?

Introduction / Context:
This question combines volume formulas for a sphere and a right circular cylinder under the principle of conservation of volume. When a solid object is melted and recast into new shapes, the total volume remains the same (ignoring any loss), which allows us to set up an equation and solve for the missing dimension, here the cylinder height.

Given Data / Assumptions:

• A solid sphere has radius R₁ = 9 cm.
• It is melted to form two solids: a sphere of radius R₂ = 6 cm and a right circular cylinder.
• The cylinder has radius r = 6 cm, equal to R₂.
• The total volume before and after melting is conserved.
• We need to find the height h of the cylinder.

Concept / Approach:
The volume of a sphere of radius R is V = (4/3) * π * R^3. The volume of a right circular cylinder with radius r and height h is V = π * r^2 * h. Conservation of volume gives:
Volume of original sphere = volume of smaller sphere + volume of cylinder.
Substituting the formulas and known radii allows us to solve for h. π appears in every term, so it can be cancelled out to simplify the algebra.

Step-by-Step Solution:
Step 1: Write volume of the original sphere: V₁ = (4/3) * π * 9^3. Step 2: Write volume of the smaller sphere: V₂ = (4/3) * π * 6^3. Step 3: Write volume of the cylinder: V_cyl = π * 6^2 * h. Step 4: Use conservation of volume: V₁ = V₂ + V_cyl. Step 5: Substitute and cancel π: (4/3) * 9^3 = (4/3) * 6^3 + 6^2 * h. Step 6: Compute powers: 9^3 = 729 and 6^3 = 216, 6^2 = 36. Step 7: Substitute: (4/3) * 729 = (4/3) * 216 + 36h. Step 8: Compute (4/3) * 729 = 4 * 243 = 972, and (4/3) * 216 = 4 * 72 = 288. Step 9: So 972 = 288 + 36h, giving 36h = 972 − 288 = 684. Step 10: Divide by 36: h = 684 / 36 = 19 cm.

Verification / Alternative Check:
We can confirm by recomputing approximate volumes with π factored out. The original sphere's volume factor is (4/3) * 729 = 972. The smaller sphere's factor is (4/3) * 216 = 288. With h = 19 and radius 6, the cylinder's factor is 6^2 * 19 = 36 * 19 = 684. Adding 288 and 684 gives 972, matching the original sphere volume factor, which verifies that h = 19 cm is consistent.

Why Other Options Are Wrong:
Heights 21 cm, 23 cm, or 25 cm would give cylinder volume factors 756, 828, or 900 respectively, which would make the total volume greater than or less than the original sphere's volume. They do not satisfy the equation V₁ = V₂ + V_cyl when the two spheres are included.

Common Pitfalls:
Students sometimes forget to cube the radii when computing sphere volumes or square the radius for the cylinder. Another common error is to forget that the cylinder has the same radius as the smaller sphere and mistakenly use 9 cm instead of 6 cm. Cancelling π early in the equation helps to simplify the calculations and reduce arithmetic mistakes.

Final Answer:
The height of the cylinder is 19 cm.