Problem #13: A tree trunk leans away from the measurer. The measurer lines up so that his/her eye and the trunk lie in the same vertical plane. Measurements are taken of: (1) the angle from eye to top of trunk, (2) eye to the bottom of the trunk, (3) distance from eye to top of trunk, (4) distance from eye to bottom of trunk, and (5) level distance from eye to trunk. With these measurements how can the measurer calculate the angle of trunk lean from the vertical?


Solution: The diagram and equations below show how the problem is solved.






In this problem, a1, a2, L1, L2, and P1P4 are measured by the measurer. The remaining quantities are calculated by the above equations. If P1P4 < L, then the solution does not work. The problem of calculating the lean when the tree leans toward the observer requires the following equations.













Summary Comments: The accompanying Excel Workbook automatically solves problems of both types. The diagram applies to type #1, i.e. lean away from measurer. ProblemSolver#1 automates type #1 and ProblemSolver#2 worksheet automates type #2.