An error of 2% in excess is made while measuring the side of a square. What is the percentage error in the calculated area of the square?

Introduction / Context:
This question examines how a small percentage error in measuring a linear dimension such as side length affects the calculated area of a square. Since area depends on the square of the side, errors do not carry over linearly. Understanding this helps in dealing with measurement errors, approximations, and sensitivity of computed quantities to errors in input data, which is important in practical applications.

Given Data / Assumptions:

• True side length of the square is taken as s.
• The side is measured 2% in excess, so the measured side becomes 102% of the true side.
• Thus, measured side = 1.02 * s.
• True area = s^2, while calculated area uses the measured side length.
• We are asked for the percentage error in the calculated area compared to the true area.

Concept / Approach:
Area of a square is proportional to the square of its side. When the side is multiplied by a factor, the area is multiplied by the square of that factor. Here the linear factor is 1.02, so the area factor is (1.02)^2. The percentage error in area is then computed by comparing this factor with 1, the factor corresponding to the true area. The difference, multiplied by 100, gives the percentage error in area.

Step-by-Step Solution:
Let the true side be s, so true area A = s^2.Measured side s' = 1.02 * s (2% in excess).Calculated area A' = (s')^2 = (1.02 * s)^2 = 1.02^2 * s^2.Compute 1.02^2 = 1.0404, so A' = 1.0404 * s^2.Percentage error in area = (1.0404 − 1) * 100% = 0.0404 * 100% = 4.04%.

Verification / Alternative check:
Take a numerical example. Suppose the true side is 10 units. The true area is 10^2 = 100 square units. If we measure the side as 2% larger, measured side = 10.2 units. The calculated area becomes 10.2^2 = 104.04 square units. The increase in area is 4.04 square units, which is 4.04% of 100, matching the computed percentage error. This confirms that the theoretical calculation is correct.

Why Other Options Are Wrong:
Option 5.04% would result from mistakenly adding an extra 1% or miscomputing 1.02^2. Option 3.96% does not correspond to any squared factor near 1.02 and is just an incorrect approximation. Option 6.04% is far too large and could come from adding 2% and 4% without justification. Option 2.02% incorrectly assumes that area error equals side error divided by 2 or some other unjustified manipulation.

Common Pitfalls:
A frequent mistake is to assume that a 2% error in side measurement will lead to the same 2% error in area, ignoring the squared relationship between side and area. Some also incorrectly double the percentage and report 4% without including the extra 0.04 part from squaring 1.02. Others may approximate too roughly or miss the step of converting the decimal difference into a percentage.

Final Answer:
The percentage error in the calculated area of the square is 4.04%.