ENTS,
Problem #4 is attached. I won't repeat it here.
I have many problems in mind that can be used to build the bank of
solved problems, but I'd rather take suggestions from those of you
who want to see certain kinds of problems addressed and solved.
Also, t he bank isn't just for me to provide input.
A good dozen or more of you out there can think up excellent
problems and present their solutions to the benefit of all.
Bob
Continued at:
http://groups.google.com/group/entstrees/browse_thread/thread/f15d03508b3a85c5?hl=en
Problem#4: A tree stands in the middle
of the south end of a flat parking lot. The parking lot slants
downward from north to south at the rate of a 1 foot drop for every
25 feet of horizontal distance. The high point of the tree is
100 vertical feet above the base and has a horizontal offset of 10
feet in the direction of north. A measurer sets out to measure the
tree with a tape and clinometer in the middle of the north end of
the parking lot. The measurer’s eye is 6 feet directly above the
ground. The measurer lays out a 100 foot baseline to the trunk of
the tree. Unbeknownst to the measurer, the clinometer is out of
calibration and reads high. Positioned at the end of the 100foot
baseline, the measurer reads the clinometer. He then makes a quick
calculation and adds the height of his eye above the base of the
tree and pronounces the tree as 114 feet in height. If the measurer
is unaware of the horizontal offset when he computes the tree’s
height, by how much is his clinometer off?
Solution: This problem needs a diagram.
We begin with the following.
Where:
D =100 = baseline to tree
k = 10 = horizontal offset of
high point
from trunk
h_{2} = (4 + 6) = 10 = height of tree below eye level
h_{1} = 90 = height of tree above eye level
The measurer begins by calculating the drop of the parking
lot as (1/25)100 = 4 feet and then adds his height to get the part
of the trunk below eye level. He does not need to use the clinometer
for the below eye level part of the tree’s height. His addition
gives him 10 feet. He can now concentrate on height above eye level.
If he announces the total height of the tree aa 114 feet, he would
have determined 104 feet were above eye level. Since by our initial
assumption, he does not know of the horizontal offset or the amount
that his clinometer is off, he obviously cannot calculate the true
height of the tree. From
the statement of the problem, can we determine the amount off his
clinometer is? First we calculate the actual angle from eye to
high point. Since the tree is 100 feet tall
by our initial assumption and 10 feet are below eye level, the
height above eye level is actually 90 feet. In addition, the
baseline to the point beneath the high crownpoint is actually
10010=90 feet (adjusting for the horizontal offset). We can
determine angle a using the arctangent function.
This is the actual angle from the measurer’s eye to the top
of the tree, but not what he reads on his clinometer. Since his
final calculation for the tree’s height is 114 feet and 10 feet are
below eye level, we have 104 feet of height that he calculates to be
above eye level. Using this information, we can calculate the angle
necessary to yield the 104foot above eye level height using his
100foot baseline.
Finally, we calculate the amount off that the clinometer
reads.
Our friend’s clinometer reads high by 1.12 degrees. Had the
clinometer read accurately, our measurer would have read 45 degrees
and the height he would have calculated for the tree using a
100foot baseline would have been 100 feet above eye level. Adding
the10 feet of height below eye level, he would have declared the
tree to be 110.0 feet in height instead of 114. His calculations
suffer from two sources of error. According to Dr. Lee Frelich, the
user fell prey to the combination of Type I and Type II measurement
errors.
Summary: This problem illustrates the
value of being able to calculate the source and impact of different
types of errors. While the mathematics involved is often very basic,
our experience shows us that the impacts of various sources of
measurement error are often not understood even by experienced
measurers.
