If the lateral surface area of a cylinder is 551.8 cm² and its height is 7 cm,…

Concept: For a right circular cylinder of radius r and height h, the lateral (curved) surface area is LSA = 2πrh, and the volume is V = πr2h. Both formulas use the same radius r, so once r is found from the LSA equation it can be substituted directly into the volume formula.

1. Solve the lateral surface area equation for the radius symbolically: LSA = 2πrh, so r = LSA ÷ (2πh).

2. Substitute this expression for r into the volume formula V = πr2h: V = π × [LSA ÷ (2πh)]2 × h = LSA2 ÷ (4πh).

3. Substitute the given values LSA = 551.8 cm², h = 7 cm, and π = 3.14: V = (551.8)2 ÷ (4 × 3.14 × 7).

4. Evaluate the numerator and denominator separately: (551.8)2 = 304483.24; 4 × 3.14 × 7 = 87.92.

5. Divide to get the volume: V = 304483.24 ÷ 87.92 ≈ 3463.19 cm³.

Cross-check: solving directly for the radius gives r = 551.8 ÷ (2 × 3.14 × 7) = 551.8 ÷ 43.96 ≈ 12.5523 cm. Substituting this rounded radius back, the lateral surface area 2 × 3.14 × 12.5523 × 7 ≈ 551.80 cm² matches the given value, and the volume 3.14 × (12.5523)2 × 7 ≈ 3463.17 cm³ agrees with the value from the algebraic formula up to the rounding of r — confirming the approximate volume is 3463.19 cm³.