For a right circular cylinder, if the radius is increased by 25%, by what percentage must the height be reduced so that the volume of the cylinder remains the same?

Introduction / Context:
This question tests understanding of how changes in dimensions affect the volume of a right circular cylinder. It is a standard percentage change and mensuration combination problem. You must correctly apply the volume formula and reason about proportional changes when one dimension increases and another must decrease to keep volume constant.

Given Data / Assumptions:

• Original cylinder has radius r and height h.
• New radius r₂ = 1.25r (a 25 percent increase).
• Volume of the cylinder must remain unchanged.
• We need the percentage decrease in height from h to h₂.
• Use volume formula V = π * r^2 * h.

Concept / Approach:
Volume of a cylinder is directly proportional to the square of the radius and directly proportional to the height. If radius is multiplied by some factor, volume is multiplied by the square of that factor, provided height stays unchanged. To keep the volume unchanged when radius increases, the height must be divided by the same square factor. Then we can express the new height as a percentage of the original and finally compute the percentage decrease.

Step-by-Step Solution:
Step 1: Original volume V₁ = π * r^2 * h. Step 2: New radius r₂ = 1.25r, so r₂^2 = (1.25)^2 * r^2 = 1.5625 * r^2. Step 3: Let new height be h₂. New volume V₂ = π * r₂^2 * h₂ = π * 1.5625 * r^2 * h₂. Step 4: For volumes to be equal, V₁ = V₂, so π * r^2 * h = π * 1.5625 * r^2 * h₂. Step 5: Cancel common terms π and r^2 to get h = 1.5625 * h₂, so h₂ = h / 1.5625. Step 6: Note that 1.5625 = 25 / 16, so h₂ = h * 16 / 25 = 0.64h. Step 7: New height is 64 percent of original, so percentage decrease = 100 − 64 = 36 percent.

Verification / Alternative check:
We can verify quickly by choosing convenient numbers. Let r = 4 units and h = 25 units. Then volume V₁ = π * 16 * 25 = 400π. New radius r₂ = 1.25 * 4 = 5, new height h₂ = 16 units (because 64 percent of 25 is 16). New volume V₂ = π * 25 * 16 = 400π, which matches the original volume. This confirms that a 36 percent reduction in height is correct when radius increases by 25 percent.

Why Other Options Are Wrong:
A 56 percent, 64 percent or 46 percent decrease would produce heights that are too small and would reduce the volume below the original value. The option 25 percent might come from mistakenly thinking the height should simply decrease by the same percentage as the radius increase, which is incorrect because volume depends on the square of the radius. Only a 36 percent decrease keeps the volume unchanged.

Common Pitfalls:
Learners often overlook that radius is squared in the volume formula and may try to match percentage change in height to the percentage change in radius. Others may apply incorrect percentage calculations when converting the factor 16 / 25 into a percentage. Always identify the exact power with which each linear dimension appears in the formula, and then convert the final multiplicative factor to a percentage change carefully.

Final Answer:
The height must be reduced by 36 percent.