Westward Ho! Distance to Yonder Mountain Mini-module CG 051108 Westward Ho! Distance to Yonder Mountain An elementary trig problem, with an excursion into error propogation Quantitative concepts and skills Trigonometry, tangent Algebra, combining equations Algebra, rearranging equations Error propagation Prepared for SSAC by Len Vacher – University of South Florida, Tampa FL © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005

Problem You are on the High Plains, heading west. In the far distance you see a high peak in the Rocky Mountains and head your wagon train straight for it. Yesterday, you carefully used an inclinometer to measure the angle between the horizontal and the line of sight from you to the top of the peak. The angle was 2.0. Today, you repeat the measurement. The angle is 2.7. You have traveled 12.0 miles since yesterday’s measurement. How many miles are you now from the peak? How many feet above you is the peak? (Assume there is no change in elevation between yesterday and today. Assume too that the Earth is flat.)

Visualizing the Problem Peak h a b a b Today Yesterday

h a b a b Visualizing the Problem Peak Today Yesterday (1) (3) (5) (6) Combine (1) and (2) Solve (4) for b (1) (3) (5) Expand (3) and rearrange Combine (1) and (5) (6) (2) (4)

Forming Your Spreadsheet Recreate this spreadsheet – So, these measurements imply that the peak is about 34 mi away and about 8500 ft above you.

Exploring Your Answer But what if your angles were slightly off? How sensitive is your calculated result to the two angles? What if there were an uncertainty of ±10 per cent in α and β ? What would be the maximum and minimum b and h ?

Suppose α were 10% larger, and β were 10% smaller. Maximum Suppose α were 10% larger, and β were 10% smaller.

Suppose α were 10% smaller, and β were 10% larger. Minimum Suppose α were 10% smaller, and β were 10% larger.

Summary So a ±10% change in the two angles propagates to more than a ± 200% change in b and more than a ± 40% change in h.

End of Module Assignment It is now a day later. You are 12 miles closer to the peak. The angle of the line of sight to the peak is now 4.15 above the horizontal. What are your new results for the distance and relative elevation of the peak? Hand in the new version of the spreadsheet of Slide 5. Assuming a ±10% uncertainty for the two angle sightings, what are the new uncertainties of the distance and relative elevation? Hand in the new version of the spreadsheet of Slide 9.