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According to an article by Thomas H. McMahon in the July 1975 issue of Scientific American, a tree’s height varies directly with the radius of the base of its trunk. He expressed this relation using the formula h = kr^2/3 where k is a constant, h is the tree’s height, and r is the tree’s radius.

Now suppose you own a stand of trees whose pulp can be used for making paper. The amount of wood pulp you can produce from a tree increases as the tree’s volume increases. The model approximates a tree without its branches as a right circular cone. The formula for the volume of the tree then becomes V = (1/3)πr^2h.

Substituting the formula for height of a tree in the formula for volume of a tree, the new formula for volume becomes

  • precalc -

    taking your formulas at face value and guessing that
    the strange symbol is supposed to be π,
    sub h = kr^2/3 into V = (1/3)πr^2h
    to get
    V = (1/3)πr^2(kr^2/3)
    = (1/9)kr^4

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