The radius of the base and the height of a right circular cone are both increased by 10 percent. By what percentage does the volume of the cone increase?

Introduction / Context: This problem illustrates how percentage changes in dimensions affect the volume of a three dimensional solid. Because volume often involves a length dimension raised to a power, a small percentage change in each dimension can produce a larger percentage change in volume. The cone is a good example, because its volume depends on the square of the radius and directly on the height.

Given Data / Assumptions:

• Original radius of base = r.
• Original height = h.
• Both radius and height are increased by 10 percent.
• Volume formula for a cone: V = (1 / 3) * π * r^2 * h.
• We are asked for the percentage increase in volume.

Concept / Approach: If a quantity x is increased by 10 percent, it is multiplied by 1.10. Here both radius and height are multiplied by 1.10. Because volume involves r^2 and h, the new volume factor relative to old volume is: New volume factor = (new r / old r)^2 * (new h / old h). So we have: New volume factor = (1.10)^2 * 1.10 = (1.10)^3. Once we compute this factor, we subtract 1 and convert to a percentage to obtain the percentage increase in volume.

Step-by-Step Solution: Step 1: Express new dimensions. New radius r1 = 1.10 * r. New height h1 = 1.10 * h. Step 2: Write original and new volumes. Original volume V = (1 / 3) * π * r^2 * h. New volume V1 = (1 / 3) * π * r1^2 * h1. Step 3: Substitute r1 and h1. V1 = (1 / 3) * π * (1.10 * r)^2 * (1.10 * h). V1 = (1 / 3) * π * (1.21 * r^2) * (1.10 * h) = (1.21 * 1.10) * (1 / 3) * π * r^2 * h. Step 4: Compute the factor 1.21 * 1.10 = 1.331. So V1 = 1.331 * V. Step 5: Percentage increase = (1.331 - 1) * 100% = 0.331 * 100% = 33.1%.

Verification / Alternative check: We can test with concrete values. Let r = 10 units and h = 10 units. Then V = (1 / 3) * π * 10^2 * 10 = (1 / 3) * π * 1000. New radius = 11, new height = 11. New volume V1 = (1 / 3) * π * 11^2 * 11 = (1 / 3) * π * 1331. The ratio V1 / V = 1331 / 1000 = 1.331, which again indicates a 33.1 percent increase.

Why Other Options Are Wrong: 33.5% and 32.1% are close but do not match the exact factor of 1.331 and represent rounding or calculation errors. 53.1% is much too high and would correspond to a much larger change in one of the dimensions. 21% suggests adding the two 10 percent increases instead of considering the cubic effect on volume.

Common Pitfalls: A common mistake is to add the percentage increases directly (10% for radius plus 10% for height) and conclude a 20 percent increase in volume, which ignores the squared dependence on radius. Another error is to compute only (1.10)^2 and forget that height also changes. Recognizing that volume scales as r^2 * h and applying the percentage factor to each dimension separately is essential to obtain the right answer.

Final Answer: The volume of the cone increases by 33.1%.