Summit Surveyor Task
Summit Surveyor
Advanced Trigonometry Performance Task
Surveyor Name:
Date:
The Challenge: Measuring Mt. Zenith
Your surveying team has been tasked with determining the exact elevation of Mt. Zenith's peak. Due to treacherous terrain, you cannot reach the base of the mountain. You must use indirect measurement from two observation stations located on a nearby plateau.
Station Alpha (\(A\))
- • Elevation: 1,250 meters
- • Angle of Elevation to Peak: \(28.4^\circ\)
Station Beta (\(B\))
- • Elevation: 1,420 meters
- • Angle of Elevation to Peak: \(35.2^\circ\)
Note: Station \(B\) is exactly 800 meters closer to the mountain's vertical axis than Station \(A\). Both stations and the peak lie in the same vertical plane.
Step 1: Visual Modeling
Construct a detailed geometric diagram of the scenario. Label all known distances, angles, and variables. Represent the mountain peak as point \(P\), and the points directly below the peak at the elevations of Station \(A\) and \(B\) as \(A'\) and \(B'\).
Survey Plotting Area
Step 2: Mathematical Formulation
Let \(h\) be the height of the peak above the elevation of Station \(A\), and let \(x\) be the horizontal distance from Station \(B\) to the mountain's vertical axis. Write two trigonometric equations that relate \(h\), \(x\), and the given data. (Hint: Remember that Station B is at a different elevation than Station A).
Equation for Station Alpha:
Equation for Station Beta:
Step 3: Finding the Height
Solve your system of equations to find the height \(h\) (the height of the peak relative to Station \(A\)). Show all algebraic steps clearly. Round your final answer for the total elevation of Mt. Zenith to the nearest meter.
Step 4: Error & Sensitivity Analysis
A. Instrument Calibration
If the angle measurement tool at Station \(B\) was found to be miscalibrated by \(+0.5^\circ\) (meaning the true angle was \(34.7^\circ\)), how would this affect your calculated height? Explain whether the calculated height would be an overestimate or an underestimate without re-calculating the entire system.
B. Curvature of the Earth
In long-distance surveying, the curvature of the Earth must be considered. Over a horizontal distance of 10km, the Earth "drops" about 7.8 meters. If the horizontal distance to the peak is approximately 8.5km, explain how this physical reality might impact your survey's accuracy.
Final Verified Elevation
Seal of Accuracy Required
_______ m
Summit Surveyor Key
Summit Surveyor
Teacher Answer Key & Scoring Rubric
MASTER COPY
Step-by-Step Solution
1. Variable Definitions
- Let \(H_P\) = Total elevation of the peak.
- Let \(h\) = Height of peak above Station A (1,250m).
- Let \(x\) = Horizontal distance from Station B to the peak's axis.
- Distance from Station A to axis = \(x + 800\).
- Vertical offset between B and A = \(1,420 - 1,250 = 170\)m.
- Height of peak above Station B = \(h - 170\).
2. The System of Equations
Station Alpha:
\(\tan(28.4^\circ) = \frac{h}{x + 800}\)
\(h = (x + 800) \tan(28.4^\circ)\)
Station Beta:
\(\tan(35.2^\circ) = \frac{h - 170}{x}\)
\(h - 170 = x \tan(35.2^\circ)\)
3. Solving for \(x\) and \(h\)
Substitute \(h = x \tan(35.2^\circ) + 170\) into Alpha's equation:
\(x \tan(35.2^\circ) + 170 = (x + 800) \tan(28.4^\circ)\)
\(x \tan(35.2^\circ) + 170 = x \tan(28.4^\circ) + 800 \tan(28.4^\circ)\)
\(x(\tan(35.2^\circ) - \tan(28.4^\circ)) = 800 \tan(28.4^\circ) - 170\)
\(x(0.7054 - 0.5407) = 800(0.5407) - 170\)
\(x(0.1647) = 432.56 - 170 = 262.56\)
\(x \approx 1,594.17\) meters
Now solve for \(h\): \(h = 1,594.17 \tan(35.2^\circ) + 170 \approx 1,294.5\) meters
Final Result
Elevation = 1,250 + 1,294.5 = 2,544.5m
Final Answer: 2,545 meters
Scoring Rubric
| Criteria | Target (4 pts) | Progressing (2 pts) | Beginning (1 pt) |
|---|
| Step 1: Modeling | Perfectly labeled diagram showing nested triangles and elevation offsets. | Diagram exists but missing one key label or offset. | Incorrect structure or missing multiple variables. |
| Step 2: Setup | Two correct trig equations properly accounting for the 170m elevation difference. | Equations correctly use tan but ignore the elevation difference between A and B. | Incorrect trig ratios or major setup errors. |
| Step 3: Execution | Algebraic substitution or elimination is correct; calculation accurate to within 5m. | Minor computational error but algebraic method is sound. | Major algebraic errors or unable to combine equations. |
| Step 4: Analysis | Correctly identifies \(+0.5^\circ\) error leads to underestimate (due to steeper slope logic) and identifies drop in elevation. |
