A solid cylinder has total surface area 462 sq cm. If its curved surface area…

For any solid (closed) cylinder of radius r and height h: Total Surface Area (TSA) = Curved Surface Area (CSA) + area of the two circular bases, i.e. TSA = 2πrh + 2πr2, and Volume = πr2h. Once the CSA is known as a fraction of the TSA, the base-area term 2πr2 is fixed, which pins down r, then h, then the volume.

1. Curved Surface Area = one-third of Total Surface Area = (1/3) × 462 sq cm = 154 sq cm.

2. Since TSA = CSA + 2πr2, the base-area term 2πr2 = 462 − 154 = 308 sq cm.

3. Taking π ≈ 22/7: 2 × (22/7) × r2 = 308, so r2 = 49, giving r = 7 cm.

4. From CSA = 2πrh: 154 = 2 × (22/7) × 7 × h = 44h, so h = 3.5 cm.

5. Volume = πr2h = (22/7) × 49 × 3.5 = 539 cubic cm.

Check: CSA + 2πr2 = 154 + 308 = 462, which equals the given total surface area, and 154 is exactly one-third of 462 — confirming the radius and height obtained are consistent.

So, using π = 22/7 — the convention the clean radius value of 7 cm confirms is intended here — the volume works out to exactly 539 cubic cm.