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True horizontal distance from slope distances

The following slope distances were measured along a chain line with a 30 m tape.

Slope distance (m) = 25.50, 24.60, 28.70, 29.50 and 18.50.

Difference of elevation between ends (m) = 2.50, 5.30, 3.35, 2.50 and 1.50.

It was noted afterwards that the tape was 2 cm too long. Find the true horizontal distance.

Method. The tape is 30.02 m, 2 cm too long, so every reading is too small. First each slope distance is corrected for tape length, then reduced to horizontal using the height difference hh:

l′=l×30.0230,D=l′2−h2l' = l\times\frac{30.02}{30},\qquad D = \sqrt{l'^{2} - h^{2}}
Slope ll (m)hh (m)Corrected slope l′l' (m)Horizontal DD (m)
25.502.5025.517025.3942
24.605.3024.616424.0391
28.703.3528.719128.5231
29.502.5029.519729.4136
18.501.5018.512318.4515
Total125.8215

(The total measured slope length is 126.80 m.)

Answer: true horizontal distance =125.822= 125.822 m.

✓ Answer checked by Claude Opus 5.5 (high effort)

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