If the radius of a right circular cylinder is decreased by 20%, by what percentage must its height be increased so that the volume of the cylinder remains unchanged?

Introduction / Context:
This question combines percentage change with the volume formula for a cylinder. The volume of a cylinder depends on both the radius and the height. When one dimension changes, the other can be adjusted so that the overall volume remains constant. Here, the radius is reduced by 20 percent, and we must determine by what percent the height should increase to compensate exactly for this reduction in base area. The solution uses algebra and the idea that the new volume must equal the original volume.

Given Data / Assumptions:
- Original cylinder has radius r and height h. - Original volume V = π * r^2 * h. - Radius is decreased by 20%, so new radius r_new = 0.8r. - Let the new height be h_new. - We require the new volume V_new to equal the original volume V.

Concept / Approach:
We start by expressing both original and new volumes in terms of r and h. The new volume is V_new = π * (r_new)^2 * h_new. With r_new = 0.8r, the square becomes 0.64r^2. Setting V_new equal to V gives an equation relating h_new to h. Once we find the factor by which the height increases, we convert that factor into a percentage increase. This approach uses proportional reasoning and avoids the need for any actual numerical value of r or h.

Step-by-Step Solution:
Step 1: Original volume is V = π * r^2 * h. Step 2: New radius is r_new = 0.8r after a 20% decrease. Step 3: New volume is V_new = π * (r_new)^2 * h_new = π * (0.8r)^2 * h_new. Step 4: Compute (0.8r)^2 = 0.64r^2, so V_new = π * 0.64r^2 * h_new. Step 5: Set V_new equal to original V for the volume to remain the same: π * 0.64r^2 * h_new = π * r^2 * h. Step 6: Cancel π and r^2 from both sides to get 0.64 * h_new = h. Step 7: Solve for h_new: h_new = h / 0.64. Step 8: Compute 1 / 0.64 = 100 / 64 = 25 / 16 = 1.5625. Step 9: Thus h_new = 1.5625h, meaning the height must be multiplied by 1.5625. Step 10: The percentage increase in height is (1.5625 − 1) * 100% = 0.5625 * 100% = 56.25%.

Verification / Alternative check:
As a quick check, choose simple values. Let r = 10 units and h = 1 unit. Then original volume is proportional to r^2 * h = 100 * 1 = 100. New radius is 8 units (20% decrease from 10), giving r_new^2 = 64. We want h_new such that 64 * h_new = 100 to keep volume constant. Solving gives h_new = 100 / 64 = 1.5625, which represents exactly a 56.25 percent increase over the original height 1. This numerical example matches the algebraic result.

Why Other Options Are Wrong:
Option 20 would simply reverse the 20 percent change, which is not correct because the radius affects volume through its square, not linearly. Option 36.25 comes from misinterpreting the change factor or making an arithmetic error when solving for h_new. Option 65 is larger than needed and would cause the new volume to exceed the original volume significantly.

Common Pitfalls:
Learners sometimes assume that a 20 percent decrease in radius requires a 20 percent increase in height, ignoring the r^2 term in the volume formula. Others may forget to square the factor 0.8, mistakenly using 0.8 instead of 0.64 and thus obtaining wrong numbers. Another common pitfall is converting the final multiplicative factor into percent incorrectly. Keeping the algebra clean and double checking the squaring and division steps helps avoid these errors.

Final Answer:
The height of the cylinder must be increased by 56.25% to keep the volume unchanged.