CELE Challenge

Master these topics to ace the Mathematics and Surveying portion of CELE!

1. Trigonometry - The Foundation

SOH-CAH-TOA (Life Saver!)

Sin th = Opposite / Hypotenuse
Cos th = Adjacent / Hypotenuse
Tan th = Opposite / Adjacent

Law of Sines

a/sin A = b/sin B = c/sin C

Used when you have: AAS, ASA, or SSA (ambiguous case)

Law of Cosines

c2 = a2 + b2 - 2ab(cos C)

Used when you have: SAS or SSS

2. Calculus Essentials

Differentiation

Power Rule: d/dx(x^n) = nx^(n-1)
Product Rule: d/dx(uv) = u(dv/dx) + v(du/dx)
Chain Rule: dy/dx = dy/du x du/dx

Applications: Max/Min problems, Rate of change

Integration

Power Rule: integral x^n dx = x^(n+1)/(n+1) + C
Definite Integral: Area under curve
Integration by Parts: integral u dv = uv - integral v du

Applications: Areas, Volumes, Work

3. Surveying Fundamentals

Tape Corrections

Temperature: Ct = aL(T - Ts)
Pull: Cp = (P - Ps)L / AE
Sag: Cs = -w2L3 / 24P2
Slope: Ch = h2/2L (for small angles)

Pacing

Pace Factor = Known Distance / Number of Paces

Unknown Distance = Number of Paces x Pace Factor

4. Leveling

Differential Leveling

HI (Height of Instrument) = Elevation + BS (Backsight)
Elevation = HI - FS (Foresight)
Check: Sum BS - Sum FS = Last Elev - First Elev

Trigonometric Leveling

V = D x tan(a) where D = horizontal distance, a = vertical angle

Elevation B = Elevation A + hi + V - hr

5. Traverse Surveying

Azimuth and Bearings

Azimuth: Measured clockwise from North (0 deg to 360 deg)
Bearing: Measured from N or S toward E or W (0 deg to 90 deg)
Back Azimuth: Forward Azimuth +/- 180 deg

Latitude and Departure

Latitude (Y) = Distance x cos(Bearing)
Departure (X) = Distance x sin(Bearing)
For a closed traverse: Sum Lat = 0, Sum Dep = 0

Error of Closure

Linear Error = sqrt[(Sum Lat)2 + (Sum Dep)2]

Relative Precision = Linear Error / Total Distance

6. Area Computation

Double Meridian Distance (DMD)

DMD1 = Departure1
DMDn = DMD(n-1) + Dep(n-1) + Depn
2A = Sum(DMD x Latitude)

Double Parallel Distance (DPD)

DPD1 = Latitude1
DPDn = DPD(n-1) + Lat(n-1) + Latn
2A = Sum(DPD x Departure)

Coordinate Method

2A = Sum(Xn x Y(n+1)) - Sum(Yn x X(n+1))

Most commonly used in modern surveying!

CELE Pro Tips - Mathematics and Surveying

- Master your scientific calculator - know all functions!
- Always check units - degrees vs radians, meters vs feet
- For traverse, always verify closure before computing area
- Memorize tape correction formulas - they appear every exam!
- Practice converting between Azimuth and Bearing

Mathematics & Surveying