Q.62 A tacheometer was set at a station and the readings were taken on two points, which were 100 m and 150 m apart and had staff intercepts of 3.98 m and 5.98 m, respectively. If an externally focusing tacheometer was used for the observation, then the values of focal length (f), stadia interval (i) and distance (d) between the objective and the vertical axis of the instrument could be
- Af = 0.3 m, i = 1.2 mm, d = 20 cm
- Bf = 0.3 m, i = 0.12 mm, d = 20 cm
- Cf = 0.25 m, i = 12 mm, d = 30 cm
- Df = 0.3 m, i = 12 mm, d = 20 cmCorrect
Solution
Concept
For an external focusing tacheometer, D = (f/i)s + (f + d). Find the constants from two observations, then check the options.
Formula
D = (f/i) × s + (f + d)
Step-by-step solution
1. Use D = K s + C, where K = f/i and C = f + d. 2. 100 = K × 3.98 + C 3. 150 = K × 5.98 + C 4. Subtract: 50 = 2K, so K = 25. 5. C = 100 - 25 × 3.98 = 100 - 99.5 = 0.5 m. 6. Check option D: f/i = 0.3/0.012 = 25 and f + d = 0.3 + 0.2 = 0.5 m. Both match.
Explanation
Two readings give two equations. Solving them gives the multiplying constant K = 25 and the additive constant C = 0.5 m. Only option D gives f/i = 25 and f + d = 0.5 m.
Why the other options are wrong
A: f/i = 0.3/0.0012 = 250, not 25. B: f/i = 0.3/0.00012 = 2500, not 25. C: f/i = 0.25/0.012 = 20.8 and f + d = 0.55 m, so neither constant matches.
Shortcut
K = ΔD/Δs = 50/2 = 25. Then pick the option with f/i = 25 and f + d = 0.5 m.
Common mistake
Using i in mm without converting to m, which gives a wrong f/i ratio.
Correct answer: D) f = 0.3 m, i = 12 mm, d = 20 cm