A line was measured by 20 m and 100 ft. chain respectively and was 12 chains in length in each case. If the 30 m chain was 0.2 m too long, find the correct length of the 100 ft. chain up to three decimal places. Take 1 m = 3.28 ft.

Problem Statement

A line was measured using both a 30 metre chain and a 100 ft chain, and the recorded length was 12 chains in each case. It was later found that the 30 m chain was actually 0.2 m too long. Determine the correct (actual) length of the 100 ft chain. Use the conversion factor 1 m = 3.28 ft and provide the answer to three decimal places.

Step-by-Step Solution

Key Information

Nominal Length of Metric Chain (L_{30m, nom}) = 30 m
Error in Metric Chain = +0.2 m
Actual Length of Metric Chain (L_{30m, act}) = 30 m + 0.2 m = 30.2 m
Nominal Length of Imperial Chain (L_{100ft, nom}) = 100 ft
Measured Length (using 30m chain) = 12 chains
Measured Length (using 100ft chain) = 12 chains
Conversion Factor = 1 m = 3.28 ft
Goal: Find the Actual Length of the 100 ft chain (L_{100ft, act}).

Step 1: Calculate True Length of the Line (using 30m chain data)

First, find the measured distance using the nominal length of the 30m chain:

Measured Distance = Number of Chains × Nominal Chain Length

Measured Distance = 12 × 30 m

Measured Distance = 360 m

Now, apply the correction formula using the actual length of the 30m chain:

True Length (L_true) = (L_{30m, act} / L_{30m, nom}) × Measured Distance

L_true = (30.2 m / 30 m) × 360 m

L_true ≈ 1.006667 × 360 m

L_true = 362.4 m

Step 2: Convert True Length to Feet

Convert the calculated true length from meters to feet using the given conversion factor.

L_{true, feet} = L_{true, meters} × Conversion Factor

L_{true, feet} = 362.4 m × 3.28 ft/m

L_{true, feet} = 1188.672 ft

Step 3: Calculate Actual Length of the 100 ft Chain

We know the true length of the line is 1188.672 ft. This distance was measured as 12 applications (chains) using the 100 ft chain.

True Length = Number of Chains × Actual Chain Length (L_{100ft, act})

Therefore, L_{100ft, act} = True Length_feet / Number of Chains

L_{100ft, act} = 1188.672 ft / 12

L_{100ft, act} = 99.056 ft

Final Result

The correct (actual) length of the 100 ft chain is 99.056 ft.

Conceptual Explanation

Why this works:

Establishing Ground Truth: The measurement made with the 30m chain, combined with the knowledge of its error (it was 30.2m long), allows us to calculate the actual, true length of the line surveyed (362.4 m or 1188.672 ft). This true length is independent of which measuring tool was used.
Relating Measurements: Since the same line was measured using the 100ft chain and recorded as 12 chain lengths, we know that 12 applications of this potentially inaccurate 100ft chain covered the true distance of 1188.672 ft.
Finding the Unknown Error: By dividing the total true distance (in feet) by the number of times the 100ft chain was laid out (12 times), we determine the average actual length of the 100ft chain during the measurement. In this case, it was 99.056 ft, meaning the 100ft chain was actually too short.
Systematic Error Correction: This process corrects for the systematic error in both measuring instruments by first using the known error of one instrument (the 30m chain) to find the true physical dimension, and then using that true dimension to calibrate or find the actual length of the second instrument (the 100ft chain).