The curved surface area of a solid cylinder closed at both ends with an height of 21 m equals one-th

For a closed cylinder with radius r and height h: curved surface area (CSA) = 2πrh, and total surface area (TSA) = 2πrh + 2πr² (the curved part plus the two circular ends).

The question says CSA is one-third of TSA, so 2πrh = ⅓(2πrh + 2πr²). Multiply both sides by 3: 6πrh = 2πrh + 2πr². Take 2πrh to the left: 4πrh = 2πr².

Divide both sides by 2πr: 2h = r. So r = 2 × 21 = 42 m.

Check with r = 42, h = 21: CSA = 2π(42)(21) = 1764π. The two ends = 2π(42²) = 3528π, so TSA = 1764π + 3528π = 5292π. And ⅓ of 5292π = 1764π, which equals the CSA. So the base radius is 42 m.