If the curved surface area of a solid cylinder is one-third of its total surface area, then what is the ratio of its diameter to its height?
4 : 1
Step 1 — Set up the condition:
For a solid cylinder, \(CSA = 2\pi r h\) and \(TSA = 2\pi r h + 2\pi r^2\). Given \(CSA = \tfrac{1}{3} TSA\):
\[3(2\pi r h) = 2\pi r h + 2\pi r^2\]
Step 2 — Solve for r in terms of h:
\[4\pi r h = 2\pi r^2 \implies r = 2h\]
Step 3 — Ratio of diameter to height:
\(d = 2r = 4h\), so \(d : h = \mathbf{4 : 1}\).
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