A solid brass cube is melted and recast as a solid cone of height h and base radius r. If the height of the cube is h, find r in terms of h.

Analyze the question: A cube of side h is melted and recast as a cone of height h and base radius r. Melting and recasting does not change the volume, so the two volumes are equal.

Concept:

• Volume of a cube = h³ (the side of the cube is h)
• Volume of a cone = (1/3)πr²h

Solution:

Set the volumes equal:

h³ = (1/3)πr²h

Divide both sides by h:

h² = (1/3)πr²

Multiply both sides by 3:

3h² = πr²

Divide by π:

r² = 3h²/π

Take the square root:

r = √(3h²/π) (equivalently, r = h√(3/π))

Checking the options:

• A. r = h is wrong, because it would need π/3 = 1.
• B. r = √(3h²/π) matches our result exactly.
• C. r = πh is wrong.
• D. r = h√(3/h) is wrong, because the factor under the root should be 3/π, not 3/h. It looks similar, so be careful.

Tip: In "melted and recast" questions, always equate the volumes. Cancel any common factor (here h) early to keep the algebra short. Also check that the π appears in the right place when comparing options.