Calculate the limiting length (m) of the offset, if the maximum allowable error in laying the offset is 2 degrees. The scale of the map is 1 cm = 100 m.

📝 Detailed Explanation: Limiting Length of an Offset

The limiting length of an offset is the maximum length an offset can have in the field so that its plotting error on the map does not exceed a specified limit. This concept connects the accuracy of fieldwork to the accuracy of the final map.

The error on the map is caused by a small angular error ($$\theta$$) made when laying out the offset in the field. This error displaces the point on the map by a small amount. The maximum displacement is limited by the sharpness of a person's vision, which is typically taken as 0.25 mm, or 0.025 cm.

Formula and Calculation

From the trigonometry of the error, we can establish the relationship:

$$ \sin\theta = \frac{\text{Displacement on Map}}{\text{Length of Offset on Map}} $$

Rearranging for the Limiting Length (L) in the field:

$$ L = \frac{\text{Allowable Plotting Error}}{\sin\theta} \times \text{Scale} $$

Given:

• Maximum allowable error in laying offset, $$\theta = 2^\circ$$
• Scale of the map = 1 cm / 100 m
• Standard allowable plotting error = 0.025 cm (a standard value representing the limit of human plotting accuracy)

Calculation:

$$ L = \frac{0.025 \text{ cm}}{\sin(2^\circ)} \times \frac{100 \text{ m}}{1 \text{ cm}} $$

$$ L = \frac{0.025 \times 100}{\sin(2^\circ)} \text{ m} $$

$$ L = \frac{2.5}{0.0349} \text{ m} $$

$$ L = 71.63 \text{ m} $$