Modelling growth of a tree

Question

(Year) 1	(Height of tree in ft) 6
2	15
3	20
4	24
5	28
6	29
7	31
8	32
9	34

Determine a logarithmic function (in the form y=a logx+d for the curve of best fit for the data. Show all of your work and explain how you determined the best model to fit the data.

Using your table and/or scatter plot, make the following predictions and then justify your predictions algebraically using the function you developed:

  a) How tall will the tree be at 30 months?

  b) How tall will the tree be in 15 years?

  c) When will the tree reach a height of 10 feet?

  d) When will the tree reach a height of 25 feet? 

Answer 1

IF this were a time pressure exam

simplest solution would be

29logx + 6

take x=1, alogx+d = 0+d = 6

d=6

then try x=10 which looks about 35 = alog10+6 = a+9 = 35, a= 29

a=29, d=6

the equation is 29logx +6

if you have time test it on a couple other x values, such as 8 or 3. It works well

if you have more time do a regression analysis, have a calculator with statistics functions

x = 30 months, that's = 30/12 = 2.5 years

29log2.5 + 6 = about 17.5 feet which is half way between 15 and 20, for x=2 and 3

x =15, 29log15+6 = 40.1 feet after 15 years

10 feet = 29logx +6

logx = 4/29

x = about 1.4 years to reach 10 feet

25 feet should be slightly more than x=4 from the table

logx =(25-6)/29 = 19/29

x = about 4.5 years to reach 25 feet

if this is a long term assignment with no real time pressure, learn how to do regression analysis

or some online websites will calculate and graph it for you