The external and internal radii of a hollow right circular cylinder are 6.75 cm and 5.25 cm respectively, and its height is 15 cm. If the metal of this hollow cylinder is melted and recast into a solid right circular cylinder whose height is half of the original height, what is the radius of the new solid cylinder?

Introduction / Context:
Here we work with volumes of solids of revolution. A hollow right circular cylinder is melted and recast into a solid cylinder. Since volume of metal is conserved in the melting and recasting process, the volume of the hollow cylinder must equal the volume of the new solid cylinder. This allows us to find the radius of the new cylinder.

Given Data / Assumptions:
• External radius of hollow cylinder Re = 6.75 cm.
• Internal radius of hollow cylinder Ri = 5.25 cm.
• Height of hollow cylinder h1 = 15 cm.
• New solid cylinder height h2 = h1 / 2 = 7.5 cm.
• Metal volume is conserved when melting and recasting.
• Use volume formulas with π constant cancelling out.

Concept / Approach:
Volume of a hollow cylinder is equal to the difference between volumes of two cylinders: Vhollow = πh1(Re2 − Ri2).
Volume of the new solid cylinder is Vsolid = πh2R2, where R is the new radius.
Equating Vhollow and Vsolid and simplifying lets us solve for R. Square radii and use simple algebra to avoid calculator dependence.

Step-by-Step Solution:
Step 1: Volume of hollow cylinder Vhollow = πh1(Re2 − Ri2). Step 2: Compute Re2 − Ri2 using the identity a2 − b2 = (a − b)(a + b). Step 3: Here a = 6.75, b = 5.25, so a − b = 1.5 and a + b = 12, giving Re2 − Ri2 = 1.5 × 12 = 18. Step 4: So Vhollow = π × 15 × 18. Step 5: Let R be the radius of the new solid cylinder. Its volume Vsolid = πh2R2 = π × 7.5 × R2. Step 6: Equate volumes: π × 15 × 18 = π × 7.5 × R2. Step 7: Cancel π and simplify: 15 × 18 = 7.5 × R2. Divide both sides by 7.5 to get R2 = (15 × 18) / 7.5 = 2 × 18 = 36. Step 8: Thus R = √36 = 6 cm.

Verification / Alternative check:
You can compute both volumes numerically. Vhollow = π × 15 × 18 = 270π. Vsolid with R = 6 gives Vsolid = π × 7.5 × 36 = 270π. Since the volumes match exactly, R = 6 cm is confirmed.

Why Other Options Are Wrong:
• 7.25 cm or 6.5 cm arise from incorrect arithmetic when evaluating Re2 − Ri2 or dividing by 7.5.
• 7 cm would need R2 = 49, which does not satisfy the volume equality.

Common Pitfalls:
A common mistake is to subtract radii directly rather than subtracting their squares in the volume formula. Another error is to forget that the new height is half the original height. Always write the complete volume equations before simplifying.

Final Answer:
The radius of the new solid cylinder is 6 cm.