Attached is the first problem that I promised. It deals with
determining trunk length of a leaning tree and is presented in a
Word document. The solution utilizes the law of cosines.
Incidentally, Paul Jost has been a good advocate of that law. Maybe
we can think up a problem or two.
Problem#1: A leaning tree is to
be measured from within the plane of the lean, i.e. the tree leans
directly toward or away from the measurer. The trunk is straight
from ground level to the branching point. The measurer has a
clinometer and laser rangefinder. Assume that the distance from the
measurer to the top of the trunk is 65 feet at an angle above the
eye of 20 degrees. Further assume that the distance from the
measurer to the bottom of the trunk is 70 feet at an angle below the
eye of 15 degrees. What is the length of the trunk segment?
Solution: The solution involves
the law of cosines. A triangle is formed from the eye to the top of
the trunk, then down along the trunk to the bottom, and then back to
the ey. The triangle must be solved for the distance down the trunk
(trunk length) using the distances from the eye to the extremities
and the intervening angle.
The figure below depicts the problem. Drawing is not to scale
D1 = distance from eye to top of
trunk = 65 ft
D2 = distance from eye to bottom of
trunk = 70 ft
angle between D1 and D2 = 20+15 = 35
D3 = distance along the trunk, i.e.
trunk length, to be determined
Law of Cosines