The radius and height of a solid right circular cylinder are both increased by 2 percent. Approximately what is the percentage increase in the volume of the cylinder?

Introduction / Context:
This question tests your understanding of percentage change in a compound situation where more than one dimension of a solid is changed. The volume of a cylinder depends on both its radius and its height. When both are changed by the same percentage, the effect on volume is not simply the sum of the percentage changes. Instead, we must use the idea of successive percentage changes or apply the formula directly with the new dimensions.

Given Data / Assumptions:
- Original radius of the cylinder is r.
- Original height of the cylinder is h.
- Radius is increased by 2 percent, so new radius is 1.02r.
- Height is also increased by 2 percent, so new height is 1.02h.
- Volume of a cylinder is V = π * r^2 * h.
- We need the approximate percentage increase in volume.

Concept / Approach:
The original volume is V = π * r^2 * h. After the changes, the new volume V new is:
V new = π * (1.02r)^2 * (1.02h)
This simplifies to V new = π * r^2 * h * (1.02^2 * 1.02) = V * (1.02^3). So the factor by which the volume increases is 1.02^3. We can either compute 1.02^3 directly or use the successive percentage change concept to estimate the overall percentage increase.

Step-by-Step Solution:
Step 1: Write original volume V = π * r^2 * h. Step 2: After a 2 percent increase, the new radius is r new = 1.02r and new height is h new = 1.02h. Step 3: New volume V new = π * (1.02r)^2 * (1.02h) = π * r^2 * h * 1.02^3. Step 4: Compute 1.02^3 = 1.02 * 1.02 * 1.02 ≈ 1.0404 * 1.02 ≈ 1.061208. Step 5: The factor 1.061208 means the volume has increased by about 6.1208 percent, which rounds to approximately 6.12 percent.

Verification / Alternative check:
You can also use the idea of successive percentage change in three steps. First, increasing r by 2 percent increases r^2 by approximately 4.04 percent (since area of a circle depends on r^2). Then increasing the height by 2 percent multiplies the volume by 1.02. So total factor is approximately 1.0404 * 1.02 ≈ 1.061208, giving about 6.12 percent increase, which matches the earlier calculation.

Why Other Options Are Wrong:
5.88 and 6.76 are close but result from rounding at intermediate steps or incorrect compounding. 3.34 and 4.04 arise if someone simply adds percentages incorrectly or only accounts for the change in radius or in radius squared without including the height change.

Common Pitfalls:
A very common mistake is to just add the percentage changes, for example 2 percent for radius and 2 percent for height, to get 4 percent, which ignores the compounding effect. Others might increase the radius squared incorrectly or forget that volume depends on r^2 and h. Always convert percentage increases into multipliers, apply them to the formula and then convert the final factor back into a percentage.

Final Answer:
The approximate percentage increase in the volume of the cylinder is 6.12 percent.