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Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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CivilAdvanced Survey
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What is the formula for finding horizontal distance if the staff is held vertical and line of sight is inclined?

A

D=kscos2θ+csin⁡θD = k s cos^{2} \theta + c \sin \thetaD=kscos2θ+csinθ

B

D=kscos⁡2θ+csin⁡θD = k s \cos 2 \theta + c \sin \thetaD=kscos2θ+csinθ

C

D=kscos2θ+ccos⁡θD = k s cos^{2} \theta + c \cos \thetaD=kscos2θ+ccosθ

D

D=kscos⁡θ2+ccos⁡θD = k s \cos \theta ^{2} + c \cos \thetaD=kscosθ2+ccosθ

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey
Option C

D=kscos2θ+ccos⁡θD = k s cos^{2} \theta + c \cos \thetaD=kscos2θ+ccosθ

Quick Summary:

When the staff is held vertical and the line of sight is inclined at an angle θ\thetaθ to the horizontal, the horizontal distance DDD between the instrument station and the staff station is given by D=kscos⁡2θ+ccos⁡θD = ks \cos²\theta + c \cos\thetaD=kscos2θ+ccosθ. Here, kkk is the multiplying constant (F/iF/iF/i), ccc is the additive constant (F+dF+dF+d), and sss is the staff intercept.

⚙️TETechnical SolutionConcept & Principle
💡 Explanation

When the staff is held vertical and the line of sight is inclined at an angle θ\thetaθ to the horizontal, the horizontal distance DDD between the instrument station and the staff station is given by D=kscos⁡2θ+ccos⁡θD = ks \cos²\theta + c \cos\thetaD=kscos2θ+ccosθ. Here, kkk is the multiplying constant (F/iF/iF/i), ccc is the additive constant (F+dF+dF+d), and sss is the staff intercept.

🔢 Key Formulas

D=kscos⁡2θ+ccos⁡θD = ks \cos²\theta + c \cos\thetaD=kscos2θ+ccosθ — Horizontal distance for vertical staff and inclined line of sight

V=12kssin⁡2θ+csin⁡θV = \frac{1}{2} ks \sin 2\theta + c \sin\thetaV=21​kssin2θ+csinθ — Vertical component for vertical staff and inclined line of sight

L=kscos⁡θ+cL = ks \cos\theta + cL=kscosθ+c — Inclined distance along line of sight

⚙️ Working Principle

The line of sight is inclined at an angle θ\thetaθ, making the staff non-perpendicular to the line of sight. The effective staff intercept perpendicular to the line of sight becomes s′=scos⁡θs' = s \cos\thetas′=scosθ. The inclined distance along the line of sight is L=ks′+c=kscos⁡θ+cL = k s' + c = ks \cos\theta + cL=ks′+c=kscosθ+c. Resolving LLL horizontally gives the horizontal distance D=Lcos⁡θ=kscos⁡2θ+ccos⁡θD = L \cos\theta = ks \cos²\theta + c \cos\thetaD=Lcosθ=kscos2θ+ccosθ.

📌 Key Points
  • ▸

    The multiplying constant k=fik = \frac{f}{i}k=if​ is typically equal to 100 in modern tacheometers.

  • ▸

    The additive constant c=f+dc = f + dc=f+d is made zero by using an anallactic telescope.

  • ▸

    When using an anallactic lens (c=0c = 0c=0), the formula simplifies to D=kscos⁡2θD = ks \cos²\thetaD=kscos2θ.

✅ Advantages
  • ▸

    Eliminates the need for direct chaining or taping over rough or hilly terrain.

  • ▸

    Allows rapid measurement of both horizontal distance and elevation differences.

❌ Disadvantages / Limitations
  • ▸

    Less precise than direct electronic distance measurement (EDM) or precise chaining.

  • ▸

    Accuracy drops significantly at high vertical angles (θ>15°\theta > 15°θ>15°).

🛠️ Applications / Uses
  • ▸

    Topographic surveying in hilly or inaccessible terrain.

  • ▸

    Contour mapping and preparation of location plans.

📄 Additional Information
  • ▸

    Option A incorrect: Uses sin⁡θ\sin\thetasinθ for the additive component instead of cos⁡θ\cos\thetacosθ.

  • ▸

    Option B incorrect: Uses cos⁡2θ\cos 2\thetacos2θ instead of cos⁡2θ\cos²\thetacos2θ and sin⁡θ\sin\thetasinθ instead of cos⁡θ\cos\thetacosθ.

  • ▸

    Option D incorrect: Places the exponent incorrectly on the angle θ\thetaθ rather than the cosine function.

📊 Diagram / Illustration
Horizontal Distance Formula in TacheometryInclined Line of Sight & Vertical StaffD = k s cos²θ + c cosθWhere:• k = Multiplying Constant (f / i, usually 100)• c = Additive Constant (f + d, usually 0 for anallactic lens)• s = Staff Intercept (S_top -S_bottom)• θ = Angle ofElevation orDepressionVertical Component: V = k s sinθ cosθ + c sinθ
✅

C is correct — The horizontal distance formula for an inclined line of sight with a vertical staff is D=kscos⁡2θ+ccos⁡θD = ks \cos²\theta + c \cos\thetaD=kscos2θ+ccosθ.

Core Concepts Used
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Tacheometry Stadia System Anallactic Lens
💡 EXAM TIP

For anallactic telescopes (c=0c = 0c=0), always use D=kscos⁡2θD = ks \cos²\thetaD=kscos2θ and V=12kssin⁡2θV = \frac{1}{2}ks \sin 2\thetaV=21​kssin2θ to save time during competitive exam calculations.

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