For Exercises 52–55, refer to the following. Maybe some trees could grow to a mile high, but they just don’t live long enough to have the chance. In this problem, we try to determine how fast the height of a tree increases. We can measure indirectly how much mass the tree adds in a year by the area of the annual tree ring added. Here are two relevant facts:

  • As you may have noticed from stumps, as a tree grows older, its annual rings get less wide. Although the width of the ring varies somewhat from year to year with the amount of rainfall and other factors, the total area of each annual ring is roughly the same over the years, meaning that the tree adds roughly the same amount of mass each year. Call that amount ; the mass of the tree is at, where is its age in years.
  • Over a large range of tree sizes and tree species, the diameter of a tree of a species is approximately proportional to the three-halves power of the height of the tree. (Different species have different constants of proportionality.) Thus, (which is shown in Exercise 52).

Now, if we assume that the bulk of the mass of the tree is in the trunk, and if we model the trunk either as a long cylinder or as a thin cone, the mass is proportional to the volume, so . Then

so . In other words, the tree grows in height as the fourth root of its age.

Question 18.85

image 55. The branching of trees is similar to the branching of circulatory systems in the bodies of animals. For similar reasons, the area of the cross section of the tree at its base scales as the three-fourths power of the tree’s mass—that is, . Assume that most of the mass is in the trunk and model the tree either as a tall cylinder or an elongated cone . Show that the diameter of a tree is approximately proportional to the three-halves power of the height—that is, .

55.

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