ENTS,
Problem #7 is attached. I will draw diagrams
for scenarios 2 and 3 if anyone wants them. We are grateful to Beth
for suggestion the inclusion of this problem into the problem set.
Bob
Problem #7: Beth Koebel sets out to
measure an oak in a ravine. However she has trouble finding a spot
where she can see both the top and bottom of the tree. She notes a
branch stub from a position where she sees the top. She sees the
same stub from a spot where the base is visible. How may Beth use
the stub to measure the full height of the tree?
Solution: The diagram below shows the
solution where the branch stub is above eye level from both vantage
points. The method of solution is to divide the tree into two
sections: top down to the stub (identified as point P in the
diagram) and from P down to the base of the oak. The equation in the
diagram consists of four terms. The first two terms are: (1) the
height from the top down to eye level, and (2) the height from point
P down to eye level. These measurements are made from vantage point
E_{1}. The second two terms are: (1) the height of P above
eye level, and (2) the height from the base up to eye level. These
measurements are made from vantage point E_{2}. The four
quantities are algebraically combined as shown in the formula, where
H stands for total height of the tree.
In this kind of problem, there are actually three scenarios
to consider. They are listed below.
Scenario #1: Point P is above eye level
from both vantage points. The equation is
The above diagram depicts this
first scenario and it is probably the most common one.
Scenario #2: Point P is below eye level
from both vantage points. The equation is
Scenario #3: Point P is below eye level
from the higher vantage points and above eye level from the lower
vantage point. The equation is
Comments: This problem illustrates a
common situation that is often encountered in the field within the
interior of a forest, i.e. no common spot where the top and bottom
can be seen. In these cases, the tree must be divided up into
portions that can be individually measured. The separate
measurements are then combined. Also, situations do occur where
positions E_{1} and E_{2} occur on the same
horizontal plane. The measurer shifts forward or backward to view
first top and then the bottom. The method of solution is the normal,
i.e. sine top – sine bottom. Diagrams similar to the one above can
be drawn to clarify each of the above scenarios.
Continued at:
