A tacheometer is setup at A and the readings on the staff at B... | Civil MCQ | Examoogle

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A tacheometer is setup at A and the readings on the staff at B are 1.m, 2.m, 3m and the inclination of line of sight is + 10°. Calculate the vertical distance. Take k = 100, c = 0 ?

$34.00\text{ m}$

$34.202\text{ m}$

$35.202\text{ m}$

$36.202\text{ m}$

Correct Answer

⚙️ TE • Technical Direct FormulaCivilAdvanced Survey

Option B

$34.202\text{ m}$

Quick Summary:

B is correct because calculating $V = 100 \times 2.0 \times \sin(10°) \cos(10°)$ gives exactly $34.202\text{ m}$ .

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📐MAMath SolutionDirect Formula

📋 Given

Staff readings = 1.0 m, 2.0 m, 3.0 m, Angle of inclination $\theta = +10°$ , Multiplying constant $k = 100$ , Additive constant $c = 0$

🔢 Formula Used

V=k⋅S⋅sin⁡(2θ)2+c⋅sin⁡(θ)V = k \cdot S \cdot \frac{\sin(2\theta)}{2} + c \cdot \sin(\theta)V=k⋅S⋅2sin(2θ)​+c⋅sin(θ)

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate Staff Intercept ( $S$ )

The staff intercept $S$ is the difference between the top hair reading ( $S_{top}$ ) and the bottom hair reading ( $S_{bottom}$ ).

S=Stop−Sbottom=3.0 m−1.0 m=2.0 mS = S_{top} - S_{bottom} = 3.0\text{ m} - 1.0\text{ m} = 2.0\text{ m}S=Stop​−Sbottom​=3.0 m−1.0 m=2.0 m

Apply Vertical Distance Formula

For an inclined line of sight, the formula for vertical distance $V$ is $V = k \cdot S \cdot \sin(\theta) \cdot \cos(\theta) + c \cdot \sin(\theta)$ , which simplifies using trigonometric identity $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$ .

V=k⋅S⋅sin⁡(2θ)2+c⋅sin⁡(θ)V = k \cdot S \cdot \frac{\sin(2\theta)}{2} + c \cdot \sin(\theta)V=k⋅S⋅2sin(2θ)​+c⋅sin(θ)

Substitute Values and Solve

Substitute $k = 100$ , $S = 2.0\text{ m}$ , $\theta = 10°$ , and $c = 0$ into the formula.

V=100×2.0×sin⁡(20°)2=100×sin⁡(20°)V = 100 \times 2.0 \times \frac{\sin(20°)}{2} = 100 \times \sin(20°)V=100×2.0×2sin(20°)​=100×sin(20°)

Final Numerical Calculation

Since $\sin(20°) \approx 0.3420201$ , we multiply by $100$ to get the vertical distance.

V=100×0.3420201=34.202 mV = 100 \times 0.3420201 = 34.202\text{ m}V=100×0.3420201=34.202 m

✅

B is correct because calculating $V = 100 \times 2.0 \times \sin(10°) \cos(10°)$ gives exactly $34.202\text{ m}$ .

Core Concepts Used

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Staff Intercept Tacheometric Constants Inclined Line of Sight

💡 EXAM TIP

In tacheometry questions, if the staff is held vertical (standard case), use $V = k S \sin(\theta) \cos(\theta) + c \sin(\theta)$ . If the staff is normal to the line of sight, the formula changes to $V = (k S + c) \sin(\theta)$ .

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