In which is following equation for the horizontal distance in tangential method of tacheometry, when both angl

Question

In which is following equation for the horizontal distance in tangential method of tacheometry, when both angles are angle of elevation.

Examoogle · Accepted Answer

The correct answer is Option A:

D = \frac{s}{\tan α_{1} - \tan α_{2}}

. In tangential tacheometry, when both measured vertical angles are angles of elevation, the upper target point lies at a height above the lower target point on the staff. By applying trigonometry to the two right-angled triangles formed with the instrument axis, the horizontal distance is given by $D = \frac{s}{ an\alpha_1 - an\alpha_2}$, where $s$ is the staff intercept ($s = h_1 - h_2$) and $\alpha_1 > \alpha_2$. Step-by-Step Solution & Concepts: In tangential tacheometry, when both measured vertical angles are angles of elevation, the upper target point lies at a height above the lower target point on the staff. By applying trigonometry to the two right-angled triangles formed with the instrument axis, the horizontal distance is given by $D = \frac{s}{ an\alpha_1 - an\alpha_2}$, where $s$ is the staff intercept ($s = h_1 - h_2$) and $\alpha_1 > \alpha_2$. Working Principle: The horizontal distance $D$ between the instrument station and the staff station is determined by observing two vertical angles ($\alpha_1$ and $\alpha_2$) to two targets on a vertical staff separated by a known distance $s$. From geometry, $h_1 = D an\alpha_1$ and $h_2 = D an\alpha_2$. Subtracting the two equations yields $s = h_1 - h_2 = D( an\alpha_1 - an\alpha_2)$, which rearranges to $D = \frac{s}{ an\alpha_1 - an\alpha_2}$. Key Formulas: $D = \frac{s}{ an\alpha_1 - an\alpha_2}$ — Horizontal distance when both vertical angles are angles of elevation; $V = D an\alpha_2 = \frac{s an\alpha_2}{ an\alpha_1 - an\alpha_2}$ — Vertical distance to the lower vane point; $ ext{RL of Staff Station} = ext{RL of HI} + V - h_2$ — Reduced level calculation using lower vane height Key Points: Tangential tacheometry is used when the telescope is not fitted with a stadia diaphragm.; Requires two pointings (observations) to two target positions (vanes) on the staff for each distance determination.; When both angles are elevation angles, $ an\alpha_1$ is greater than $ an\alpha_2$, so the denominator is the difference $( an\alpha_1 - an\alpha_2)$. Advantages: Can be performed using an ordinary transit theodolite without stadia hairs.; Useful over long sights where stadia wires cannot be clearly read. Disadvantages: Requires two vertical angle measurements per reading, increasing operational time.; Any movement of the staff between the two readings introduces measurement error. Applications: Distance and elevation measurement in rugged or hilly terrain.; Surveying work when specialized tacheometric instruments are unavailable. Additional Info: Option B ($D = \frac{s}{ an\alpha_1 + an\alpha_2}$) applies when one angle is an angle of elevation and the other is an angle of depression.. Option C ($D = \frac{s an\alpha_2}{ an\alpha_1 - an\alpha_2}$) represents the vertical height $V = h_2$ to the lower target, not the horizontal distance $D$. Answer: A is correct — The horizontal distance formula when both observed vertical angles are angles of elevation is $D = \frac{s}{ an\alpha_1 - an\alpha_2}$. Exam Tip: Remember the denominator rule for tangential tacheometry: if both angles are on the same side of the horizontal (both elevation or both depression), subtract the tangents; if on opposite sides (one elevation, one depression), add the tangents.

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Option B: D=stanα1+tanα2

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Option C: D=stanα2tanα1-tanα2

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Option D: None of these

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