Determining an equation of an apple tree and growth.

Question

Year	Height (ft)
1	7
2	17
3	20
4	24
5	27
6	29
7	32
8	33
9	34

2. Using Desmos, construct a scatter plot with the year on the x-axis (as the independent variable) and the height on the y-axis (as the dependent variable). Save your scatter plot as you will need it for steps 4 and 5.

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

4. 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? 

5. Using your scatter plot from step 2 graph your logarithmic function you determined in step 3 on the same graph. Reflect upon the accuracy of your function. Are there any other transformations you could have applied to your logarithmic function to improve its fit?

DO NOT USE ANY CALCULUS METHODS OR PROGRAMS

Answer 1

Unfortunately I am not able to help with parts 2 and 5 of this question as I cannot bring Desmos output into this box. Please contact me if you continue to have trouble with those aspects of the question.

Part 3 asks use to build a logarithmic model for our data of the form y=a+blog(x). We do this using the least squares fitting method that gives us the following two equations.

b=n(∑ylog(x)-∑y∑log(x))/(∑log(x)²-(∑log(x))²

a=(∑y-b∑log(x))/n

In order to use these equations we calculate the following sums:

1. ∑y=7+17+20+24+27+29+32+33+34=45
2. ∑log(x)=log(1)+log(2)+log(3)+log(4)+log(5)+log(6)+log(7)+log(8)+log(9)=5.5598
3. ∑ ylog(x)=7log(1)+17log(2)+20log(3)+24log(4)+27log(5)+29log(6)+32log(7)+33log(8)+34log(9)=159.837
4. ∑ log(x)²=log(1)²+log(2)²+log(3)²+log(4)²+log(5)²+log(6)²+log(7)²+log(8)²+log(9)²=4.215

We can then substitute the appropriate values into our equation for b and solve for its value

b=n(∑ylog(x)-∑y∑log(x))/(∑log(x)²-(∑log(x))²=9*((159.837)-(45)(5.5598))/(4.215-(5.5598)²)=9*(-90.352)/-26.6964)=30.46

We can then substitute the appropriate values into our equation for a and solve for its value

a=(∑y-b∑log(x))/n=(223-(30.46)(5.5598))/9=5.96

Therefore our model is y=5.96+30.46log(x)

In part 4 we make predictions based on the table, since I don't have the scatterplot, and then compare them to our functions output.

4a)Since 30 months is halfway between 2 and three years we would guess from the table a tree height of 18.5 feet, halfway between 17ft and 20 ft. Our model tells us y=5.96+30.46log(2.5)=32.83... which means there is quite a bit of error in the early years of growth within our model

4b)The table does not go out to fifteen years, but we can see that in the last three years it grew by one foot per year so we can guess that it will grow six feet in the sixteen years after our table... or to a height of 40 feet. Our model tells us y=5.96+30.46log(15)=37.47. This is definitely closer to our estimate, but a little high

4c)We expect the height of the tree to reach 10 feet between years 1 & 2, so 1.5 years based on the table. We can reverse our model and solve for x, giving us the equation x=10^(y-5.96)/30 Using substitution we get x=1.36 years, very close to our estimate

4d)We expect the height of our tree to reach 25 feet between years 4 & 5, so 4.5 years based on the table. If we use the reversed equation above and substitute we get x=4.2 years, again very close to our estimate.