Problem 19

Problem #19: Some problems that I will present are more academic, with seemingly no immediate practical application. This may be one of them.

Suppose you are standing on the edge of a vertical ledge shooting a tree across a ravine. Your measuring partner is directly beneath you at the base of the ledge. A vertical line through your eye passes through your partnerís eye, i.e, he two of you are in absolute vertical alignment. Each of you shoot the tree and announce its height. The results differ. Who is right? You have reason to doubt the calibration of your partnerís clinometer. You know your clinometer and rangefinder are very accurate. Can you determine what the angle to the crown your partner should have gotten by way of a derived formula? Yes, you can. You first determine the vertical distance between your eye position and that of your partnerís. The distance forms one leg of a triangle to be explained in the solution.              

Solution: The Excel attachment shows the solution to the problem. A plane triangle is formed from your eye to the crown-point back down to your partnerís eye and then vertically up to yours. It is formation of this triangle that is key to the solution of the problem. As with most other problems, Iíve included an Excel workbook with a ďProblemSolverĒ spreadsheet. You can use the ProblemSolver to test out different scenarios. 

The mathematical process used to solve the problem employs both the law of sines and the law of cosines. The law of cosines is first used to calculate the distance from your partnerís eye to the crown-point. You know the distance from your eye to the crown-point and the distance from your eye down to your partnerís eye. Then the law of sines is used to calculate the angle between the vertical line between your and your partnerís eyes and the line from your partnerís eye to the crown-point. The angle registered by your partnerís clinometer up to the crown-point is 90 degrees minus this last angle. It is a little difficult to describe in words. The diagram illustrates the angles.