Problem #8 is attached. It represents a brief change
of pace and hopefully will stimulate interest in volumes. I'll go
back to more conventional problems for #9.
Problem #8: A white pine has a base
radius of 1.75 feet and a total height of 145 feet. From outward
appearances, the trunk looks very conical in shape. A cone with a
base radius of 1.75 feet and a height of 145 feet would have a
volume of 465 cubic feet. Above the base, the measurer is able to
get diameter measurements using a Dendrometer for heights up to 105
feet before limbs and foliage obscures the trunk above that height.
How high up the trunk must the measurer go to reach 97% of the trunk
volume, assuming a conical shape from bottom to top with the radius
of the base and the height as given above?
Solution: The diagram and definitions
below shows the cone shape and variables needed to solve the
R = radius of base of cone
H = total height of cone
V = total volume of cone
P = specified percentage of total volume
calculated from base upward
VP = actual volume corresponding to
p = percentage of the remainder of the total
h = height from top down at which percentage p
L = height above base at which percentage P is
Substitutions & calculations:
H = 145
R = 1.75
P = 0.97
Comments: This problem illustrates how
much of the volume of a cone is concentrated in the lower portion.
For the tree of this problem, at 29.91 feet above the base, 50% of
the volume of the entire trunk has been reached. This is not
intuitively obvious, but it is true so long as the shape of the tree
in the example is that of a cone. A more challenging problem is to
calculate heights and percentages where volumes are reached when the
shape is not conical.