The following table relates ages, in years, of oak trees to their base trunk circumferences, in
inches.
1. Determine an exponential function, y = abx, that fits the data. Enter this model into y = with 3-
digit accuracy.
2. Convert your exponential model in part b to base e. Use C as the independent variable and A
as the dependent variable.
3. Convert your answer to part c to the simplest logarithmic form possible.
4. What would be the circumferences of oak trees planted half a century ago and a century ago?
5. Why might it make more sense to use the base circumference of a tree to predict its age rather
than the base diameter?circumference (C)69.51112.5141516.51717.518age
(A)8101214161820222426
Solution
1)
I entered these coordinates into
http://www.xuru.org/rt/ExpR.asp#CopyPaste :
(6,8)
(9.5,10)
(11,12)
(12.5,14)
(14,16)
(15,18)
(16.5,20)
(17,22)
(17.5,24)
(18,26)
to get A = 3.9684e^(0.1015C) or A = 3.9684*(1.10683)^C
as the equivalent exponential regression equation
2)
In terms of e : A = 3.9684e^(0.1015C)

3)
A / 3.9684 = e^(0.1015C)
Ln(A / 3.9684) = 0.1015C
C = (1/0.1015) * Ln(A / 3.9684)
C = 9.8522 * Ln(A / 3.9684) ---> This is equivalenmt Log equation
4)
Half a century ago, A = 50 :
C = 9.8522 * Ln(50 / 3.9684) = 24.96 units
A century ago, A = 100 :
C = 9.8522 * Ln(100 / 3.9684) = 31.79 units
5)
The diameter is not visually accessible to us when looking at a tree, but the circumference is. So,
it makes more sense to use the circumference rather than the diameter. And also, with passing
time, nothing would be expected tp happen within a tree, but there would be effects seen to the
outside surface, i.e the circumference of the tree