Introduction / Context:
This question is from the topic of mensuration and specifically deals with the surface area of a cube. Cubes are common in aptitude exams because the formulas are simple yet many students confuse the effect of percentage changes in edge length on volume and surface area. Here we are not asked about the new surface area itself, but about the percentage increase in surface area when every edge of the cube is increased by 50 percent. The key idea is that surface area is proportional to the square of the edge length, so any percentage change in side leads to a different amplified percentage change in area.
Given Data / Assumptions:
- Original cube edge length = a units (symbolic).
- Each edge is increased by 50 percent.
- New cube edge length = 1.5 * a.
- We need the percentage increase in total surface area.
- Total surface area of a cube with edge a is 6 * a^2.
Concept / Approach:
For a cube with edge length a, total surface area S is given by:
S = 6 * a^2
If the edge length is multiplied by a factor k, then the new surface area S new becomes:
S new = 6 * (k * a)^2 = 6 * k^2 * a^2
Hence, the ratio of new surface area to original surface area is k^2. From this, we can compute the percentage increase as:
Percentage increase = (k^2 - 1) * 100 percent
Here the factor k is 1.5 since the sides are increased by 50 percent.
Step-by-Step Solution:
Step 1: Let original edge be a.
Original surface area S = 6 * a^2.
Step 2: Compute new edge after 50 percent increase.
New edge = a + 0.5 * a = 1.5 * a.
Step 3: Compute new surface area.
S new = 6 * (1.5 * a)^2
S new = 6 * 2.25 * a^2
S new = 13.5 * a^2.
Step 4: Compute increase and percentage.
Increase in area = S new - S = 13.5 * a^2 - 6 * a^2 = 7.5 * a^2.
Percentage increase = (7.5 * a^2 / 6 * a^2) * 100 percent
Percentage increase = (7.5 / 6) * 100 percent = 1.25 * 100 percent = 125 percent.
Thus the cube surface area increases by 125 percent.
Verification / Alternative check:
Take a simple numerical value for the edge, for example a = 2 units. Then original area S = 6 * 2^2 = 24. New edge is 3 units, so new area S new = 6 * 3^2 = 54. The increase in area is 54 - 24 = 30. Percentage increase is (30 / 24) * 100 percent = 125 percent. This matches our algebraic result and verifies the calculation.
Why Other Options Are Wrong:
Option B (150 percent) would correspond to k^2 equal to 2.5, which does not match 1.5 squared. Option C (175 percent) and Option D (110 percent) also do not match the required factor. Option E (100 percent) would mean the area doubled, which occurs when side is increased by about 41.42 percent, not by 50 percent.
Common Pitfalls:
The most common mistake is to assume that if the edge increases by 50 percent, the surface area also increases by 50 percent. This ignores the square relationship between side and area. Another mistake is to mistakenly use the volume relation (which is proportional to the cube of the edge) when the question clearly asks about surface area. Always recall: for a cube, surface area is proportional to a^2 and volume is proportional to a^3.
Final Answer:
The total surface area of the cube increases by
125% when each edge is increased by 50 percent.
Discussion & Comments