Make a table of values for the functions you find in Part A.
A
a
y=120(1.1)^t and y=154(1.06)^t
B
b
Graph:
Comparison: See solution.
Practice makes perfect
a
We will model the basal area of a tree using an exponential growth function.
y= a(1+ r)^t
Here, a is the initial basal area of the tree, r is the rate of growth, and t is time in years.
Writing a Function for Tree A
We see that the t-values have a common difference of 1, and that the A-values have a common ratio of 1.1.
The initial value of the basal area is 120 square inches, and the growth factor is 1.1. Using the exponential growth function form, we can write a function that represents the basal area of Tree A.
y= a( 1+r)^t ⇒ y= 120( 1.1)^t
Writing a Function for Tree B
We know that the initial value of the basal area of Tree B is 154 square inches and the growth rate is 6 %, or 0.06. Using the exponential growth function form, we can write a function that represents the basal area of Tree B.
y= a(1+ r)^t ⇒ y= 154(1+ 0.06)^t
Let's add the terms.
y= 154( 1.06)^t
b
We will draw the graphs of the functions y= 120(1.1)^t and y=154(1.06)^t. We will use the table in Part A to draw the function y=120(1.1)^t. Let's make a table of values for the other function.
x
154(1.06)^t
y=154(1.06)^t
0
154(1.06)^0
154
1
154(1.06)^1
≈ 163.2
2
154(1.06)^2
≈ 173.0
3
154(1.06)^3
≈ 183.4
4
154(1.06)^4
≈ 194.4
We now know the following.
The points ( 0, 120), ( 1, 132), ( 2, 145.2), ( 3, 159.7), and ( 7, 175.7) are on the graph of the function y=120(1.1)^t.
The points ( 0, 154), ( 1, 163.2), ( 2, 173.0), ( 3, 183.4), and ( 7, 194.4) are on the graph of the function y=154(1.06)^t.
Let's now plot and connect them with smooth curves.
We see that for the first 6 years Tree B has a greater basal area than Tree A. After about 7 years, Tree A has a greater basal area than Tree B. The basal area of Tree A grows faster.
