Step 1: Understanding the Question:
A measurement error occurs when measuring the side length of a square. The measured side is 5% larger than the true side. We need to calculate the resulting net percentage error in the calculated area of the square.
Step 2: Key Formula or Approach:
Area of a square depends on the square of its side ($A = s^2$). When a quantity is a product of two variables that each undergo a percentage change, the net percentage change formula is $x + y + \frac{xy}{100}$. Here, $x$ and $y$ are both $+5%$.
Step 3: Detailed Explanation:
Method 1: Successive Percentage Change Formula
Since Area = Side $\times$ Side, the side is effectively multiplied twice.
The percentage error in the side is $+5%$. So, $x = 5$ and $y = 5$.
Net percentage error = $5 + 5 + \frac{5 \times 5}{100}$.
$= 10 + \frac{25}{100} = 10 + 0.25 = 10.25%$.
Method 2: Using Arbitrary Values
Let the true side length of the square be $100$ units for simplicity.
The true area of this square would be $100 \times 100 = 10,000$ square units.
An error of 5% in excess means the measured side is 5% longer than the true side.
Measured side length = $100 + 5% \text{ of } 100 = 105$ units.
The calculated area based on this flawed measurement = $105 \times 105$.
Let's compute $105^2$: $105 \times 105 = 11,025$ square units.
The absolute error in the area is the difference between the calculated area and the true area.
Absolute error = $11,025 - 10,000 = 1,025$ square units.
The percentage error is the absolute error divided by the true area, multiplied by 100.
Percentage Error = $\left( \frac{1025}{10000} \right) \times 100 = 10.25%$.
Step 4: Final Answer:
The error % in the calculated area is 10.25%.