ENTS,
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 partner’s 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 partner’s 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
partner’s. 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 partner’s
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, I’ve included
an Excel workbook with a “ProblemSolver” 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 partner’s 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
partner’s eye. Then the law of sines is used to calculate the angle between the
vertical line between your and your partner’s eyes and the line from your
partner’s eye to the crown-point. The angle registered by your partner’s
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.
Bob