**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, a_{1}, a_{2}, L_{1},
L_{2}, and P_{1}P_{4} are measured by the measurer. The
remaining quantities are calculated by the above equations. If P_{1}P_{4}
< 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.

measure/problems/Problem_13.xls