ENTS,
Problem #5 is presented in the two
attachments. The Word document presents the problem with the
solution and summary comments. The spreadsheet is a handy dandy
tool to compute the answer to problems of the type described
automatically . As with other spreadsheets, the green cells are for
user input.
With this submission, the bank now contains 5
solved problems. I am willing to continue presenting problems if at
least a few of you want them, but before sending more problems, I
would appreciate a show of hands from those of you who want t he
process to continue. I'd like to test the demand for this kind of
online help. If just a few of you, say 5 or 6 , want the process to
continue , I'll be happy to oblige .
Bob
Problem 5 xls
Continued at:
http://groups.google.com/group/entstrees/browse_thread/thread/c9466ee141badad3?hl=en#
Problem #5: A measurer using a tape and
clinometer wants to limit the error in calculating tree height
attributable to the horizontal offset of the crownpoint relative to
the trunk. The measurer has been told that if the crownpoint,
trunk, and measurer’s eye are all in the same vertical plane and
then the measurer circles around 90 degrees from that vertical
plane, so that the new vertical plane that includes the trunk and
measurer’s eye is at 90 degrees relative to the first plane, then
the error in the height calculation will be reduced to a minimum.
However, the measurer knows that circumstances do not always permit
the measurer to circle a full 90 degrees and still retain visibility
of the crownpoint. How can the measurer develop a mathematical
process to compute the crownpoint offset error for different
combinations of crown height, distance to trunk, and angle between
the vertical planes.
Solution: The following diagram shows
the variables and their relationships. The left diagram is a top
down view. P_{1} is a point on the trunk, P_{2} is
the crownpoint, and P_{3} is the measurer’s eye. The right
view shows the height of the crownpoint relative to the measurer.
Definitions:
a = angle between direction of crownpoint offset and
measurer’s eye. Trunk is
the vertex of the triangle
b = angle of crownpoint being measured relative to
measurer’s eye.
D = distance from measurer to trunk.
d = horizontal offset distance from trunk to crownpoint.
n = distance from P_{2} to the line D which runs from
P_{1} and P_{3}. The line n
is at a right angle to D. The lines n and D are horizontal.
m = distance to P1 from intersection of n and D. The lines n,
m, and d form a
right triangle.
D_{1} = distance from end of m to P_{3.} D =
D_{1} + m.
S = horizontal distance from P_{3} to P_{2}.
H = height of crownpoint above measurer’s eye.
E = error in height from using D as baseline instead of S
H_{O} = S  D
From these definitions we can solve the problem through the
following sequence of steps:
With this sequence of step, we can determine the error from
initial values of a,b,d, and D. The following table illustrates the
procedure.
a

d

D

b

n

m

D_{1}

S

H_{1}

H_{2}

E

H_{O}

45.00

10.00

100.00

37.00

7.07

7.07

92.93

93.20

70.23

75.36

5.13

6.80

90.00

10.00

100.00

37.00

10.00

0.00

100.00

100.50

75.73

75.36

0.38

0.50

0.00

10.00

100.00

37.00

0.00

10.00

90.00

90.00

67.82

75.36

7.54

10.00

In the table, a, d, D, and b are the knowns. The values of
other variables are computed. The measurer can analyze the error for
different value sets. From the example, the 90 degree scenario gives
almost the exact value. If the measurer were close to a very tall
tree with a big horizontal offset, the error would increase
dramatically. Errors will generally be under 10 feet for the most
exaggerated situations. However, errors in the over 10foot range
can occur.
Summary Discussion: This problem
illustrates how a known source of error with clinometer and tape can
be minimized without employing crownpoint crosstriangulation, a
method if done correctly will eliminate the horizontal offset error
entirely. In actual field practice, visibility of different points
in the crown from different vantage points often eliminates
crownpoint crosstriangulation as a viable field technique for
accurately measuring the height of a tree.
