Difficulty: Easy
Correct Answer: 44%
Explanation:
Introduction / Context:
This is a geometry and percentage question involving a cube. It tests your ability to understand how a percentage change in the side length of a cube affects its surface area, which depends on the square of the side length.
Given Data / Assumptions:
Concept / Approach:
The total surface area of a cube is 6 * s^2. When the side length changes, the surface area changes in proportion to the square of the scale factor applied to the side. If the side becomes k times the original, the surface area becomes k^2 times the original. Therefore, we compute the side length multiplier and then square it.
Step-by-Step Solution:
Step 1: Side length is increased by 20%, which is 20/100 = 0.20.Step 2: New side length = s * (1 + 0.20) = 1.20s.Step 3: Original surface area A original = 6s^2.Step 4: New surface area A new = 6 * (1.20s)^2.Step 5: Compute (1.20s)^2 = 1.44s^2.Step 6: Therefore A new = 6 * 1.44s^2 = 1.44 * 6s^2.Step 7: So the new surface area is 1.44 times the original surface area.Step 8: Percentage increase = (1.44 − 1.00) * 100% = 0.44 * 100% = 44%.
Verification / Alternative check:
Take a simple numerical example. Let s = 10 units. Original surface area = 6 * 10^2 = 600. New side = 12, new surface area = 6 * 12^2 = 6 * 144 = 864. Increase = 864 − 600 = 264. Percentage increase = 264 / 600 * 100 = 44%, which confirms the result.
Why Other Options Are Wrong:
Forty percent is the increase in linear dimension misapplied to area. One hundred forty four percent would mean an impossible more than doubling of the area calculation for this case. Seventy two point eight percent does not correspond to the square of 1.20.
Common Pitfalls:
A common mistake is to assume that the percentage increase in surface area is the same as the percentage increase in side length. However, surface area depends on the square of the side, so the effect is larger.
Final Answer:
The surface area of the cube increases by 44%.
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