One side of a square is measured as 16.7 cm to an accuracy of 0.1 cm. What is the percentage error in area?

Explanation

To find the percentage error in the area of a square when one side is measured, we can follow these steps:

### Step 1: Understand the formula for the area of a square
The area \( A \) of a square is given by the formula:
\[ A = \text{side}^2 \]

### Step 2: Identify the given values
We are given:
- The length of one side of the square \( s = 16.7 \, \text{cm} \)
- The accuracy of the measurement \( \Delta s = 0.1 \, \text{cm} \)

### Step 3: Calculate the area of the square
Using the formula for the area:
\[ A = s^2 = (16.7 \, \text{cm})^2 = 278.89 \, \text{cm}^2 \]

### Step 4: Determine the error in the area
The error in the area \( \Delta A \) can be calculated using the formula:
\[ \frac{\Delta A}{A} = 2 \frac{\Delta s}{s} \]
This is because the area is proportional to the square of the side length.

### Step 5: Substitute the known values into the error formula
Substituting \( \Delta s = 0.1 \, \text{cm} \) and \( s = 16.7 \, \text{cm} \):
\[ \frac{\Delta A}{A} = 2 \cdot \frac{0.1}{16.7} \]

### Step 6: Calculate the fraction
Calculating the fraction:
\[ \frac{0.1}{16.7} \approx 0.005988 \]
Now, multiply by 2:
\[ \frac{\Delta A}{A} \approx 2 \cdot 0.005988 \approx 0.011976 \]

### Step 7: Convert to percentage
To find the percentage error, multiply by 100:
\[ \text{Percentage Error} = \frac{\Delta A}{A} \times 100 \approx 0.011976 \times 100 \approx 1.1976\% \]

### Step 8: Round the result
Rounding to two decimal places gives:
\[ \text{Percentage Error} \approx 1.2\% \]

### Final Answer
The percentage error in the area of the square is approximately **1.2%**.

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