# Problem Statement A surveyor measures a distance of 300 m using a chain that is .1 m too long per 30 m. Calculate the **actual distance**. --- # Step-by Solution ## 1. Understanding the Problem - **Measured**: 300 m (using the chain) - **Error in chain**: Chain is .1 m too long for every 30 m marked. - **Goal**: Find the actual (true) distance. --- ## 2. Calculate the Correction per Meter - For every 30 m, the chain is .1 m too long. - **Correct length of chain**: 30 m (nominal) - **Actual length of chain**: 30.1 m (because it's too long) ### How does this affect the measurement? - When the surveyor measures 30 m on the ground using the chain, they're actually covering 30.1 m in reality. #### The true/actual distance for every 30 m measured is 30.1 m. --- ## 3. Find the Correction Factor Let: - \( L_m \) = Measured length - \( L_a \) = Actual length - \( l \) = Nominal length of chain (30 m) - \( c \) = Correction per chain (.1 m) The actual length corresponding to measured length is: \[ L_a = L_m \times \frac{l + c}{l} \] Here, for every 30 m chain, the actual length is 30.1 m. --- ## 4. Calculate for 300 m Measured distance = 300 m \[ L_a = 300 \times \frac{30.1}{30} \] \[ L_a = 300 \times 1.00333 \] \[ L_a = 301 \text{ m} \] --- # Final Answer **The actual distance is \(\boxed{301~\text{m}}\).** --- ## Summary Table | Measured Distance | Nominal Chain Length | Actual Chain Length | Correction per 30 m | Actual Distance | |-------------------|---------------------|---------------------|---------------------|----------------| | 300 m | 30 m | 30.1 m | .1 m | 301 m |