Cylindrical pillar – find diameter : height ratio from CSA and volume: A cylindrical pillar has curved surface area 264 m^2 and volume 924 m^3. Find the ratio of its diameter to its height.

Introduction / Context:
Given CSA and volume of a cylinder, we can express height h in terms of r using CSA, then substitute into the volume formula to solve for r. Once r is known, diameter : height follows directly. This combines two standard cylinder relations.

Given Data / Assumptions:

• CSA = 2πrh = 264
• Volume V = πr^2h = 924
• Unknowns: r (m) and h (m)

Concept / Approach:
From CSA: h = 264 / (2πr) = 132 / (πr). Substitute into V to eliminate h and solve r. Then compute h explicitly and take (diameter):(height) = (2r):h.

Step-by-Step Solution:
h = 132 / (πr)V = πr^2h = πr^2 * (132 / (πr)) = 132r = 924 → r = 924 / 132 = 7 mh = 132 / (π * 7) ≈ 132 / 21.991 ≈ 6 mDiameter = 2r = 14 mRatio (diameter : height) ≈ 14 : 6 = 7 : 3

Verification / Alternative check:
Plug r = 7, h ≈ 6: CSA = 2π * 7 * 6 = 84π ≈ 263.9 ≈ 264; Volume = π * 49 * 6 ≈ 924, both consistent.

Why Other Options Are Wrong:
3:7 and 6:7 invert or mismatch the derived ratio; 7:6 assumes h = 12 rather than ≈ 6; 14:5 overstates the diameter relative to height.

Common Pitfalls:
Forgetting that CSA uses 2πrh; solving volume first without eliminating h; rounding too early.

Final Answer:
7 : 3