A right circular cylinder of maximum volume is cut from a solid cube of edge…

Concept: For a right circular cylinder cut from a solid cube, the circular cross-section must fit inside a square face and the cylinder's height cannot exceed the cube's edge. Since the volume V = πr²h increases with both r and h, the volume is largest when both are pushed to their maximum possible values at the same time — that is, when the diameter equals the cube's edge (giving the largest possible radius) and the height equals the cube's edge. So for a cube of edge a, the maximum-volume cylinder has diameter = a, radius r = a/2, and height h = a.

Application:

1. Cube edge a = 14 cm, so radius r = a / 2 = 14 / 2 = 7 cm.

2. Height of the cylinder h = a = 14 cm.

3. Total surface area of a cylinder = 2πr(r + h).

4. Substitute the values: TSA = 2 × (22/7) × 7 × (7 + 14) = 2 × (22/7) × 7 × 21.

5. The 7 in the denominator cancels with the r = 7: TSA = 2 × 22 × 21 = 924 cm².

Cross-check: Compute the curved and base areas separately: curved surface area = 2πrh = 2 × (22/7) × 7 × 14 = 616 cm²; area of the two circular bases = 2 × πr² = 2 × (22/7) × 49 = 308 cm². Adding them gives 616 + 308 = 924 cm², confirming the total surface area of 924 cm².