The edge length of a cube is increased by 10%.\nBy what percentage does the volume of the cube increase as a result of this change?

Introduction / Context:
This question tests spatial and percentage reasoning by connecting the linear dimension of a cube (its edge length) with its volume. When a side length changes by a certain percentage, the volume, which depends on the cube of the side length, changes by a different percentage. Understanding this relationship is vital in geometry and in many real world applications involving scaling.

Given Data / Assumptions:

• The original edge length of the cube is assumed to be a units.
• The edge length is increased by 10%.
• The volume of a cube is equal to side^3.
• We must find the percentage increase in volume after the side length increases by 10%.

Concept / Approach:
If the original side length is a, the original volume is a^3. When the side length is increased by 10%, the new side length becomes a * 1.10. The new volume is then (1.10 * a)^3. To find the percentage change in volume, we compare the new volume with the original volume and compute (new volume - original volume) / original volume * 100. This problem demonstrates that a linear increase leads to a larger multiplicative change in volume because of the cubic relationship.

Step-by-Step Solution:
Step 1: Let the original edge length be a, so original volume V1 = a^3.Step 2: After a 10% increase, the new edge length is a * (1 + 10/100) = 1.10 * a.Step 3: New volume V2 = (1.10 * a)^3.Step 4: Expand the cube: V2 = 1.10^3 * a^3.Step 5: Compute 1.10^3 = 1.10 * 1.10 * 1.10 = 1.331.Step 6: So V2 = 1.331 * a^3.Step 7: Increase in volume = V2 - V1 = 1.331 * a^3 - 1 * a^3 = 0.331 * a^3.Step 8: Percentage increase in volume = (0.331 * a^3 / a^3) * 100 = 0.331 * 100 = 33.1 percent.

Verification / Alternative check:
We can verify using a simple numerical value. Take a = 10 units. Original volume V1 = 10^3 = 1,000 cubic units. New side length after 10% increase is 11 units. New volume V2 = 11^3 = 1,331 cubic units. The increase in volume is 1,331 - 1,000 = 331 cubic units. As a percentage of the original volume, this is 331 / 1,000 * 100 = 33.1 percent. This matches the algebraic solution exactly.

Why Other Options Are Wrong:
30 percent and 31.3 percent underestimate the true cubic effect of the side increase and therefore give too small a percentage change in volume.
136.1 percent is far too large and would correspond to more than doubling the volume, which does not happen with a modest 10% side increase.
10 percent corresponds only to the change in side length, not to the change in volume, and so it is not the correct answer for this question.

Common Pitfalls:
Many students incorrectly assume that the volume increases by the same percentage as the side length, that is 10%, or mistake the volume increase as proportional to the square of the change instead of the cube. Another error is to forget to cube the factor 1.10 when computing the new volume. The safe approach is always to express the new dimension as a multiplier of the old one, then raise this multiplier to the appropriate power in the formula and finally compute the resulting percentage change relative to the original value.

Final Answer:
The volume of the cube increases by 33.1 percent when its edge length is increased by 10 percent.