The curved surface area of a cylindrical pillar is 264 square metres and its volume is 924 cubic metres. What is the ratio of the diameter of the base to the height of the pillar?

Introduction / Context:
Here we have a cylindrical pillar for which both the curved surface area and the volume are given. The task is to find the ratio of the diameter of the base to the height. This problem tests the ability to use two standard formulas of a cylinder together and eliminate variables to find a ratio, rather than any single value. It is a good example of using algebra with geometric formulas.

Given Data / Assumptions:

• Curved surface area (CSA) = 264 m^2.
• Volume (V) = 924 m^3.
• Radius of the base = r metres.
• Height of the cylinder = h metres.
• We must find the ratio (diameter : height) = (2r : h).

Concept / Approach:
For a cylinder, we use:
CSA = 2 * pi * r * h Volume V = pi * r^2 * h We are given both CSA and V, so we can form a ratio CSA / V to eliminate pi and h. This will give a relationship involving only r. From that, we solve for r and then find h from either formula. Finally, we compute the ratio 2r : h and simplify it to the smallest whole-number ratio.

Step-by-Step Solution:
CSA = 2 * pi * r * h = 264. V = pi * r^2 * h = 924. Compute CSA / V: (2 * pi * r * h) / (pi * r^2 * h) = 264 / 924. Left side simplifies to 2 / r. So 2 / r = 264 / 924. Simplify 264 / 924: divide both by 12 gives 22 / 77 = 2 / 7. Thus 2 / r = 2 / 7, so r = 7 m. Now use volume: 924 = pi * 7^2 * h = pi * 49 * h. So h = 924 / (49 * pi). Using CSA instead is easier: 264 = 2 * pi * 7 * h = 14 * pi * h. Hence h = 264 / (14 * pi). With pi ≈ 22 / 7, h ≈ 6 m. Diameter = 2r = 14 m, height ≈ 6 m, so ratio 14 : 6 = 7 : 3.

Verification / Alternative check:
Take r = 7 and h = 6. Then CSA = 2 * pi * 7 * 6 = 84pi, which with pi ≈ 22 / 7 gives 84 * 22 / 7 = 264 m^2. Volume = pi * 49 * 6 = 294pi, which is 294 * 22 / 7 = 924 m^3. Both values match the given data, confirming r = 7 and h = 6 are correct and, therefore, the ratio is valid.

Why Other Options Are Wrong:
Ratios like 3 : 7 or 6 : 7 either invert or misrepresent the relationship between diameter and height. For instance, 3 : 7 would imply a very slender cylinder, which does not satisfy the given CSA and volume simultaneously. Ratios 7 : 6 or 5 : 3 correspond to other inconsistent combinations of r and h that fail to satisfy both formulas at once.

Common Pitfalls:
Students sometimes forget that the question asks for diameter, not radius, and incorrectly give r : h instead of 2r : h. Another typical error is not simplifying the fraction 264 / 924 correctly, which leads to an incorrect r. It is also easy to make algebraic mistakes when forming the ratio CSA / V, so careful simplification is important.

Final Answer:
The required ratio of diameter to height of the cylindrical pillar is 7 : 3.