CELE Surveying (Geomatics) — Advanced and Geodetic SurveyingExam Answer Templates

Marks

1

Topic

Stadia Measurement

Difficulty

easy

Template Id

T1

Examiner Tip

For 1-mark definition questions, one well-constructed sentence that contains the method name, the measured quantity, and the instrumental feature will always earn the mark. Do not write a paragraph.

Model Answer

Stadia surveying is an indirect tacheometric method of measuring horizontal distance and elevation difference by reading a stadia intercept on a leveling rod using the upper and lower stadia hairs of a transit or theodolite telescope.

Question Type

very_short_answer

Answer Structure

• One complete sentence: name the method, state what is measured (distance/elevation), state the instrument feature used (stadia hairs) [1 mark]

Scoring Breakdown

Common Mark Deductions

• Saying 'measures distance with a tape' — this confuses direct with indirect methods.
• Omitting the role of the stadia hairs — just saying 'uses a telescope' is insufficient.
• Describing leveling instead of distance measurement.

Key Phrases To Include

• stadia hairs
• stadia intercept
• indirect distance measurement
• tacheometry
• horizontal distance

Marks

2

Topic

Stadia Measurement

Difficulty

easy

Template Id

T2

Examiner Tip

Examiners specifically check whether you can define s correctly as the intercept (top minus bottom stadia reading), not a single reading. Mentioning K = 100 as a typical value signals practical knowledge.

Model Answer

The stadia distance formula for a horizontal sight is: D = Ks + C where: D = horizontal distance from instrument to rod (m) K = stadia interval factor (dimensionless; typically 100 for most instruments) s = stadia intercept — the difference between upper and lower stadia hair readings on the rod (m) C = additive constant (m); approximately 0 for internal-focusing telescopes

Question Type

very_short_answer

Answer Structure

• Line 1: Write the formula D = Ks + C [1 mark]
• Line 2–5: Correctly define all four terms (D, K, s, C) with units [1 mark]

Scoring Breakdown

Common Mark Deductions

• Writing D = Ks without the additive constant C.
• Defining s as a single hair reading instead of the difference between upper and lower readings.
• Omitting units for D, s, and C.

Key Phrases To Include

• D = Ks + C
• stadia interval factor K
• stadia intercept s
• additive constant C
• K = 100
• C ≈ 0 for internal-focusing

Marks

2

Topic

Triangulation and Trilateration

Difficulty

easy

Template Id

T3

Examiner Tip

The word 'differentiate' requires a direct comparison. State both definitions and end with an explicit contrast statement. Examiners reward the contrast sentence because it proves you understand the relationship, not just the isolated definitions.

Model Answer

Triangulation is a control survey method in which the angles of a network of connected triangles are measured, while only a few distances (baselines) are directly measured; the remaining side lengths are computed using the Law of Sines. Trilateration, on the other hand, is a method in which all the sides (distances) of the triangle network are measured directly using EDM or GNSS, and the angles are then computed from the measured distances. Key distinction: Triangulation measures angles (distance computed); trilateration measures distances (angles computed).

Question Type

short_answer

Answer Structure

• Sentence 1: Define triangulation — angles measured, distances computed via Law of Sines [1 mark]
• Sentence 2: Define trilateration — distances measured, angles computed; mention EDM/GNSS [1 mark]

Scoring Breakdown

Common Mark Deductions

• Reversing the definitions — saying triangulation measures distances.
• Omitting the Law of Sines connection for triangulation.
• Writing only one definition without contrasting the two.

Key Phrases To Include

• angles measured
• Law of Sines
• baseline
• distances measured
• EDM
• control survey

Marks

2

Topic

Stadia Measurement

Difficulty

easy

Template Id

T4

Examiner Tip

Always show the subtraction s = upper − lower as a separate step. Examiners cannot award the first mark if they cannot see where your s value came from.

Model Answer

Given: Upper stadia hair reading = 1.755 m Lower stadia hair reading = 0.955 m K = 100, C = 0, horizontal sight Step 1 — Compute stadia intercept: s = 1.755 − 0.955 = 0.800 m Step 2 — Apply stadia formula: D = Ks + C D = 100(0.800) + 0 ∴ D = 80.0 m

Question Type

numerical

Answer Structure

• Step 1: List given data and compute s = upper − lower [1 mark]
• Step 2: Substitute into D = Ks + C and state final answer with unit [1 mark]

Scoring Breakdown

Common Mark Deductions

• Using only one stadia hair reading as s instead of the difference.
• Forgetting to write the unit 'm' in the final answer.
• Multiplying K by each hair reading separately instead of by the intercept.

Key Phrases To Include

• s = upper − lower
• s = 0.800 m
• D = Ks + C
• D = 80.0 m

Marks

3

Topic

Geodetic vs Plane Surveying

Difficulty

medium

Template Id

T5

Examiner Tip

For 3-mark comparison questions, the safest structure is Definition A (1 mark) + Definition B (1 mark) + Application/Condition (1 mark). The third mark is often the 'when' or 'why' — do not omit it.

Model Answer

Plane surveying treats the Earth's surface as a flat plane, ignoring its curvature. Distances, angles, and areas are computed using plane geometry and plane trigonometry. It is sufficiently accurate for small areas (typically less than 250 km² or sights shorter than about 10–15 km). Geodetic surveying accounts for the Earth's curvature and, for precise work, its actual shape as an oblate spheroid (ellipsoid). Positions are referenced to a geodetic datum (latitude, longitude, ellipsoidal height), and computations use spherical or ellipsoidal trigonometry. It is the basis for national control networks and mapping. Geodetic methods are necessary when: • The survey area spans large distances (> 15 km) where curvature errors become significant; • High-precision positioning is required (e.g., national control points, boundary demarcation between provinces); • Long geodetic sights are involved, requiring curvature-refraction correction: h_cr = 0.0675D² m (D in km).

Question Type

short_answer

Answer Structure

• Paragraph 1: Define plane surveying — flat Earth assumption, plane geometry, small areas [1 mark]
• Paragraph 2: Define geodetic surveying — accounts for curvature/ellipsoid, spherical trig, datum reference [1 mark]
• Bullet list: State at least two specific situations requiring geodetic methods, include curvature formula [1 mark]

Scoring Breakdown

Common Mark Deductions

• Saying plane surveying is 'less accurate' without explaining the curvature omission.
• Omitting any mention of datum or coordinate reference for geodetic surveying.
• Failing to give specific conditions or distances where geodetic methods are needed.

Key Phrases To Include

• flat plane
• Earth's curvature
• oblate spheroid
• geodetic datum
• latitude and longitude
• spherical trigonometry
• h_cr = 0.0675D²
• national control network

Marks

3

Topic

Stadia Measurement — Inclined Sight

Difficulty

medium

Template Id

T6

Examiner Tip

Write cos²α explicitly — do not write cos(α²). The superscript applies to the entire cosine function. Many students lose marks by writing the wrong formula. PRC examiners specifically test cos²α vs cos α.

Model Answer

Given: s = 1.20 m, K = 100, C = 0, α = 8° Step 1 — Horizontal distance: D_H = Ks cos²α D_H = 100(1.20) cos²(8°) cos(8°) = 0.99027 cos²(8°) = 0.98063 D_H = 120 × 0.98063 ∴ D_H = 117.68 m ≈ 117.7 m Step 2 — Vertical component: V = ½ Ks sin(2α) 2α = 16° sin(16°) = 0.27564 V = ½ × 100 × 1.20 × 0.27564 V = 60 × 0.27564 ∴ V = 16.54 m ≈ 16.5 m

Question Type

numerical

Answer Structure

• Step 1: Write D_H = Ks cos²α, substitute values, compute cos²(8°), state D_H [1.5 marks — formula + correct answer]
• Step 2: Write V = ½Ks sin(2α), compute 2α = 16°, substitute, state V [1.5 marks — formula + correct answer]

Scoring Breakdown

Common Mark Deductions

• Using cos α instead of cos²α for D_H — this is the most frequent error; loses the formula mark.
• Forgetting to double the angle: using sin(α) instead of sin(2α) for V.
• Rounding cos²α prematurely leading to significant error in D_H.

Key Phrases To Include

• D_H = Ks cos²α
• V = ½Ks sin(2α)
• cos²(8°)
• sin(16°)
• 2α

Marks

3

Topic

Geodetic Surveying — Curvature and Refraction

Difficulty

medium

Template Id

T7

Examiner Tip

The unit of D in this formula is km — this is the most common trap in board exams. Write 'D in km' explicitly when stating the formula to signal to the examiner that you are aware of this requirement.

Model Answer

The curvature-refraction correction (h_cr) is the combined adjustment applied to a leveled or geodetic sight to account for: (1) the Earth's curvature — which causes the line of sight to deviate above the curved Earth surface, and (2) atmospheric refraction — which bends the line of sight slightly downward, partially offsetting the curvature. Formula: h_cr = 0.0675 D² (m) where D is the sight distance in kilometres (km). Computation for D = 5 km: h_cr = 0.0675 × (5)² h_cr = 0.0675 × 25 ∴ h_cr = 1.6875 m ≈ 1.69 m This correction is added to the rod reading (or subtracted from the elevation) to obtain the true geodetic elevation difference.

Question Type

numerical

Answer Structure

• Sentence 1–2: Define curvature-refraction correction — Earth curvature + atmospheric refraction components [1 mark]
• Line 3: Write formula h_cr = 0.0675D² (D in km) [1 mark]
• Lines 4–6: Substitute D = 5 km, compute h_cr = 1.69 m with unit [1 mark]

Scoring Breakdown

Common Mark Deductions

• Using D in metres instead of kilometres — gives an answer 10⁶ times too small.
• Omitting the refraction component and saying it is only a curvature correction.
• Not stating units of h_cr in the final answer.

Key Phrases To Include

• h_cr = 0.0675D²
• D in kilometres
• Earth curvature
• atmospheric refraction
• 1.69 m
• geodetic leveling

Marks

3

Topic

Stadia Measurement — Inclined Sight Theory

Difficulty

hard

Template Id

T8

Examiner Tip

Concept-based marks in PRC exams require causal reasoning, not just formula recall. Examiners look for the phrase 'projection' or 'component' to award marks for the cos²α explanation.

Model Answer

For an inclined sight at vertical angle α, the stadia formulas are: D_H = Ks cos²α (horizontal distance) V = ½ Ks sin(2α) (vertical component) where K is the stadia interval factor, s is the stadia intercept read on the rod, and α is the vertical angle. The reason cos²α appears (not cos α) is a two-step geometric reduction: Step 1: The slope distance D_slope = Ks cos α — this is because when the sight is inclined, the effective stadia intercept projected onto a plane perpendicular to the line of sight is s cos α, so the slope distance = K(s cos α). Step 2: To get the horizontal projection of the slope distance: D_H = D_slope × cos α = Ks cos α × cos α = Ks cos²α Thus, one cos α comes from projecting the rod intercept onto the inclined direction, and the second cos α comes from projecting the slope distance onto the horizontal plane.

Question Type

short_answer

Answer Structure

• Lines 1–2: State both formulas D_H = Ks cos²α and V = ½Ks sin(2α) [1 mark]
• Lines 3–4: Explain first cos α — projection of intercept onto line of sight (slope distance = Ks cos α) [1 mark]
• Lines 5–6: Explain second cos α — horizontal projection of slope distance, yielding cos²α total [1 mark]

Scoring Breakdown

Common Mark Deductions

• Stating only the formulas without the geometric explanation of cos²α.
• Explaining only one of the two cosine factors.
• Confusing D_H with slope distance and not distinguishing them.

Key Phrases To Include

• D_H = Ks cos²α
• V = ½Ks sin(2α)
• slope distance
• projection onto horizontal
• two-step reduction
• vertical angle α

Marks

5

Topic

Triangulation — Law of Sines

Difficulty

medium

Template Id

T9

Examiner Tip

In triangulation Law of Sines problems, always draw the triangle first, label all three vertices, and write OPPOSITE pairs (side opposite to angle). AC is opposite angle B; AB is opposite angle C. This visual check prevents the most common pairing error.

Model Answer

Given: Baseline AB = 1,500 m Angle BAC (at A) = 62°30' = 62.500° Angle ABC (at B) = 71°15' = 71.250° Step 1 — Convert angles to decimal degrees: Angle A = 62°30' = 62 + 30/60 = 62.500° Angle B = 71°15' = 71 + 15/60 = 71.250° Step 2 — Compute angle C by angle sum of triangle: Angle C = 180° − 62.500° − 71.250° Angle C = 46.250° Step 3 — Apply the Law of Sines: AB / sin C = AC / sin B 1500 / sin(46.250°) = AC / sin(71.250°) Step 4 — Evaluate sines: sin(46.250°) = 0.72236 sin(71.250°) = 0.94693 Step 5 — Solve for AC: AC = AB × sin(B) / sin(C) AC = 1500 × 0.94693 / 0.72236 AC = 1420.395 / 0.72236 ∴ AC = 1,966.4 m

Question Type

numerical

Answer Structure

• Step 1: Convert DMS angles to decimal degrees [1 mark]
• Step 2: Compute angle C = 180° − A − B [1 mark]
• Step 3: Write Law of Sines ratio correctly: AB/sin C = AC/sin B [1 mark]
• Step 4: Substitute and evaluate sine values [1 mark]
• Step 5: Correct final answer AC = 1,966.4 m with unit [1 mark]

Scoring Breakdown

Common Mark Deductions

• Using angle A instead of angle B in the Law of Sines for AC — this mixes up which angle is opposite which side.
• Not converting DMS to decimal degrees, leading to incorrect sine evaluation.
• Forgetting to compute angle C and instead assigning an arbitrary value.
• Setting up the Law of Sines inverted (sin C / AB = sin B / AC).

Key Phrases To Include

• Law of Sines
• AB/sin C = AC/sin B
• angle sum = 180°
• DMS to decimal degrees
• sin(46.250°)
• sin(71.250°)

Marks

5

Topic

Stadia Measurement — Elevation Computation

Difficulty

hard

Template Id

T10

Examiner Tip

The elevation formula Elev_B = Elev_A + HI + V − r is a complete package: HI raises you to the instrument height, V elevates the sight line, and r brings you back down to the rod foot. Drawing the vertical profile sketch will make this formula intuitive and error-free.

Model Answer

Given: K = 100, C = 0, s = 0.650 m, α = +12° HI = 1.45 m, rod reading (middle hair) r = 1.80 m Elevation of A = 215.00 m Step 1 — Horizontal distance: D_H = Ks cos²α cos(12°) = 0.97815 cos²(12°) = 0.95677 D_H = 100 × 0.650 × 0.95677 D_H = 65.0 × 0.95677 ∴ D_H = 62.19 m Step 2 — Vertical component: V = ½ Ks sin(2α) 2α = 24° sin(24°) = 0.40674 V = ½ × 100 × 0.650 × 0.40674 V = 32.5 × 0.40674 ∴ V = 13.22 m Step 3 — Elevation of B: Elev_B = Elev_A + HI + V − r Elev_B = 215.00 + 1.45 + 13.22 − 1.80 ∴ Elev_B = 227.87 m

Question Type

numerical

Answer Structure

• Step 1: D_H = Ks cos²α — show cos²(12°), compute D_H = 62.19 m [1 mark for formula; 1 mark for correct answer]
• Step 2: V = ½Ks sin(2α) — show sin(24°), compute V = 13.22 m [1 mark]
• Step 3: Elev_B = Elev_A + HI + V − r — substitute all values, compute 227.87 m [2 marks: 1 for correct formula arrangement, 1 for correct final elevation]

Scoring Breakdown

Common Mark Deductions

• Using Elev_B = Elev_A + HI + V + r (adding rod reading instead of subtracting) — sign error on r.
• Forgetting to include HI in the elevation computation.
• Using α = 12° directly in sin(2α) without doubling to 24°.
• Using cos α instead of cos²α for D_H.

Key Phrases To Include

• D_H = Ks cos²α
• V = ½Ks sin(2α)
• Elev_B = Elev_A + HI + V − r
• cos²(12°)
• sin(24°)
• HI = 1.45 m
• rod reading

Marks

1

Topic

Geodetic Surveying — Datum

Difficulty

easy

Template Id

T11

Examiner Tip

Board exams frequently ask for Philippine-specific applications. PRS92 is the current national geodetic datum — memorize this. WGS84 is global (used by GPS); PRS92 is localized for the Philippines.

Model Answer

A geodetic datum is a reference system that defines the position of the coordinate origin and orientation of a coordinate system relative to the Earth's ellipsoid, used as a basis for geographic position (latitude, longitude, height). The Philippines uses the Philippine Reference System 1992 (PRS92), based on the GRS80 ellipsoid.

Question Type

very_short_answer

Answer Structure

• One sentence: define datum as a reference ellipsoid-based coordinate system AND name PRS92 [1 mark]

Scoring Breakdown

Common Mark Deductions

• Naming WGS84 as the Philippine datum — WGS84 is a global datum; PRS92 is the national datum.
• Defining datum only as a 'reference point' without mention of ellipsoid or coordinate system.
• Naming the old Luzon Datum without noting the current PRS92.

Key Phrases To Include

• reference system
• ellipsoid
• latitude and longitude
• PRS92
• Philippine Reference System 1992

Marks

2

Topic

Triangulation — Baseline

Difficulty

easy

Template Id

T12

Examiner Tip

Two-mark short answers need two clear ideas. One mark is for the definition; one mark is for the significance. Write two distinct sentences — do not combine them into one run-on sentence that may obscure the second point.

Model Answer

A baseline in triangulation is a precisely measured horizontal distance between two control stations that serves as the known side from which all other distances in the triangulation network are computed using the Law of Sines. Importance: The accuracy of the entire triangulation network depends directly on the precision of the baseline measurement. An error in the baseline propagates to all computed sides; therefore, baselines are measured with the highest available precision using invar tapes, EDM, or GNSS, and are often verified by measuring at both ends of the network (a base-check line).

Question Type

short_answer

Answer Structure

• Sentence 1: Define baseline — precisely measured distance, starting side of the network [1 mark]
• Sentence 2: State importance — all network distances depend on it; errors propagate; measured with high precision [1 mark]

Scoring Breakdown

Common Mark Deductions

• Saying the baseline is just 'any starting distance' without emphasizing precision.
• Omitting the error propagation significance.
• Confusing baseline with a traverse leg.

Key Phrases To Include

• precisely measured
• known side
• Law of Sines
• control stations
• error propagation
• invar tape or EDM

Marks

3

Topic

Geodetic Surveying — Curvature and Refraction

Difficulty

medium

Template Id

T13

Examiner Tip

When a question asks 'is this significant?', always answer quantitatively — state the computed value and compare it to a standard (closure tolerance, rod reading precision). A qualitative answer alone rarely earns the comparison mark.

Model Answer

Given: D = 8 km Step 1 — Apply curvature-refraction formula: h_cr = 0.0675 × D² h_cr = 0.0675 × (8)² h_cr = 0.0675 × 64 ∴ h_cr = 4.32 m Step 2 — Significance in plane surveying: A correction of 4.32 m over 8 km is very significant — it exceeds the allowable closure error of virtually all engineering leveling surveys. In plane surveying, this correction is ignored because plane surveys are restricted to small areas (short sights) where h_cr remains negligibly small. For example, at D = 1 km, h_cr = 0.0675 m = 67.5 mm — already measurable and significant for precise leveling. Conclusion: For sights exceeding 1–2 km, the curvature-refraction correction must be applied; it cannot be ignored in geodetic or long-distance leveling.

Question Type

numerical

Answer Structure

• Step 1: Apply h_cr = 0.0675D², substitute D = 8 km, compute h_cr = 4.32 m [1 mark]
• Step 2: State 4.32 m is large/significant; explain why plane surveying ignores it (short sights only) [1 mark]
• Step 3: Give a quantitative context (e.g., h_cr at 1 km = 0.068 m) or state the threshold [1 mark]

Scoring Breakdown

Common Mark Deductions

• Computing h_cr = 0.0675 × 8 = 0.54 m (forgetting to square D).
• Saying the correction is 'negligible' without justification.
• Not answering the second part about plane surveying significance.

Key Phrases To Include

• h_cr = 0.0675D²
• D = 8 km
• 4.32 m
• significant
• plane surveying ignores curvature
• short sights
• geodetic leveling

Marks

5

Topic

Stadia Measurement — Field Procedure

Difficulty

hard

Template Id

T14

Examiner Tip

Five-mark procedure questions reward numbered, logical steps over flowing paragraphs. Use numbered headings (1, 2, 3...) and include at least one formula in the computation step. Examiners scan for completeness across all major phases: setup → orientation → observation → computation → QC.

Model Answer

Procedure for Stadia Topographic Survey from a Single Instrument Setup: 1. INSTRUMENT SETUP (1 mark area) Set up and level the transit or theodolite at the instrument station. Measure the height of instrument (HI) above the station mark using a tape. Record the elevation of the instrument station from control data. 2. ORIENTATION (1 mark area) Sight a reference azimuth mark or backsight station. Set the horizontal circle to the reference azimuth (or to zero for relative bearings). This allows horizontal angles to all detail points to be referenced from a common direction. 3. STADIA READINGS AT EACH DETAIL POINT (1 mark area) For each point to be located: (a) Direct the telescope to the rod held vertically at the detail point. (b) Read and record the upper stadia hair (u), lower stadia hair (l), and middle hair (r) readings. (c) Read and record the vertical angle α (+ for elevation, − for depression). (d) Read and record the horizontal angle to the point. Compute stadia intercept: s = u − l. 4. COMPUTATION OF POSITION AND ELEVATION (1 mark area) For each detail point: D_H = Ks cos²α (horizontal distance) V = ½Ks sin(2α) (vertical component) Elev_point = Elev_station + HI + V − r Plot horizontal distance and direction to locate each point on the plan. 5. FIELD NOTES AND QUALITY CHECKS (1 mark area) Record all readings systematically in a stadia field book. Perform a closure check by re-observing the first detail point at the end of the session. Ensure stadia intercepts are positive (lower hair < upper hair). Unusual readings should be re-observed before moving to the next point.

Question Type

long_answer

Answer Structure

• Point 1: Instrument setup — level instrument, measure HI, record station elevation [1 mark]
• Point 2: Orientation — backsight reference, set horizontal circle [1 mark]
• Point 3: Stadia readings — upper, lower, middle hair readings + vertical and horizontal angles [1 mark]
• Point 4: Computations — D_H, V, and elevation formula for each point [1 mark]
• Point 5: Field notes, quality checks, closure verification [1 mark]

Scoring Breakdown

Common Mark Deductions

• Omitting the orientation/backsight step — a survey without reference direction is unusable.
• Not mentioning HI measurement in setup.
• Listing only the formulas without describing the physical field procedure.
• Omitting the quality check or closure verification step.

Key Phrases To Include

• height of instrument (HI)
• vertical angle
• horizontal angle
• stadia intercept s = u − l
• D_H = Ks cos²α
• Elev = Elev_A + HI + V − r
• backsight orientation
• closure check

Marks

5

Topic

Stadia Measurement — Depression Angle

Difficulty

hard

Template Id

T15

Examiner Tip

Depression angles are the PRC board exam's most-tested stadia scenario. The moment you see a negative vertical angle or the word 'depression', circle it, write V = NEGATIVE, and carry that sign all the way through the elevation formula. A sign error costs you the last two marks.

Model Answer

Given: K = 100, C = 0 s = 0.480 m α = −6°30' = −6.500° (depression angle) HI = 1.55 m, r = 2.10 m Elev_A = 340.25 m Step 1 — Convert angle to decimal: α = −6°30' = −(6 + 30/60) = −6.500° |α| = 6.500° Step 2 — Horizontal distance: D_H = Ks cos²α cos(6.500°) = 0.99356 cos²(6.500°) = 0.98715 D_H = 100 × 0.480 × 0.98715 D_H = 48.0 × 0.98715 ∴ D_H = 47.38 m Step 3 — Vertical component (negative for depression): V = ½ Ks sin(2α) [depression: V is negative] 2|α| = 13.000° sin(13.000°) = 0.22495 |V| = ½ × 100 × 0.480 × 0.22495 |V| = 24.0 × 0.22495 = 5.40 m ∴ V = −5.40 m (depression) Step 4 — Elevation of rod station: Elev_B = Elev_A + HI + V − r Elev_B = 340.25 + 1.55 + (−5.40) − 2.10 Elev_B = 340.25 + 1.55 − 5.40 − 2.10 ∴ Elev_B = 334.30 m

Question Type

numerical

Answer Structure

• Step 1: Convert DMS to decimal degrees, recognize depression (negative V) [1 mark]
• Step 2: D_H = Ks cos²α — compute cos²(6.5°), D_H = 47.38 m [1 mark]
• Step 3: V = ½Ks sin(2α) — V = −5.40 m (depression, negative sign) [1 mark]
• Step 4: Elevation formula — correct sign for V, correct final answer 334.30 m [2 marks: 1 for formula setup, 1 for correct answer]

Scoring Breakdown

Common Mark Deductions

• Treating depression angle as elevation (positive V) — the single most common error in this problem type.
• Not doubling the angle when computing sin(2α); using sin(6.5°) instead of sin(13°).
• Adding r instead of subtracting in the elevation formula.
• Omitting DMS-to-decimal conversion leading to incorrect trigonometric values.

Key Phrases To Include

• depression angle
• V is negative
• D_H = Ks cos²α
• V = ½Ks sin(2α)
• Elev_B = Elev_A + HI + V − r
• cos²(6.5°)
• sin(13°)