Problem Statement
The area of a certain field was measured with a 30 m chain and found to be 5000 sq. m. It was afterwards detected that the chain used was 10 cm too short. What is the true area of the field?
Step-by-Step Solution
Key Information
- Nominal Length of Chain (Lnom) = 30 m
- Error in Chain = -10 cm = -0.1 m (Note: “too short” means the error is negative)
- Actual Length of Chain (Lact) = 30 m – 0.1 m = 29.9 m
- Measured Area (Ameasured) = 5000 sq. m
- Goal: Find the True Area (Atrue).
Step 1: Calculate True Area
For area measurements, the correction factor based on the chain length must be squared because the error affects the measurement in two dimensions (length and width).
The formula for true area is:
Atrue = ( Lact / Lnom )² × Ameasured
Substitute the known values:
Atrue = ( 29.9 m / 30 m )² × 5000 sq. m
Atrue = ( 0.996667 )² × 5000 sq. m
Atrue ≈ 0.993344 × 5000 sq. m
Atrue ≈ 4966.72 sq. m
Final Result
The true area of the field is approximately 4966.72 sq. m.
Conceptual Explanation
Why the Correction Factor is Squared for Area:
- Two-Dimensional Effect: Area is calculated from two linear dimensions (like length × width). If the measuring tool (the chain) has a linear error (it’s too short or too long), this error affects *both* dimensions used to calculate the area.
- Correction Factor Application: The linear correction factor is (Actual Length / Nominal Length), which is (Lact / Lnom). When calculating the true area, this factor must be applied to both dimensions involved. Therefore, the overall correction factor for area becomes (Lact / Lnom) × (Lact / Lnom) = (Lact / Lnom)².
- Impact of Error: In this case, the chain was shorter (29.9m instead of 30m). This means each linear measurement taken was slightly less than recorded, leading to an overestimation of the area when calculated directly. Applying the squared correction factor ((29.9/30)²) correctly scales down the measured area to find the smaller true area. Conversely, if the chain were too long, the true area would be larger than the measured area.
- Volume Correction: Similarly, if calculating volume from measurements made with an incorrect chain, the correction factor would need to be cubed: (Lact / Lnom)³.
