The table shows the linear relationship between the average height in feet of trees on a tree farm and the number of years since the trees were planted.
Number of Years Since the
Trees Were Planted
1
4
9
16
Average Height (ft)
12
39
84
147
What is the rate of change of the average height in feet of the trees on the farm with respect to the number of years since the trees were planted?
9 ft/yr
27 ft/yr
3 ft/yr
12 ft/yr
To find the rate of change, we need to calculate the slope of the linear relationship between the number of years and the average height of the trees.
First, we'll calculate the change in average height for each change in the number of years:
- The change in average height from 1 year to 4 years is 39 - 12 = 27 feet.
- The change in average height from 4 years to 9 years is 84 - 39 = 45 feet.
- The change in average height from 9 years to 16 years is 147 - 84 = 63 feet.
Next, we'll calculate the corresponding changes in the number of years:
- The change in number of years from 1 to 4 is 4 - 1 = 3 years.
- The change in number of years from 4 to 9 is 9 - 4 = 5 years.
- The change in number of years from 9 to 16 is 16 - 9 = 7 years.
Finally, we'll calculate the average rate of change by dividing the change in average height by the corresponding change in number of years:
- The average rate of change from 1 to 4 years is 27/3 = 9 ft/yr.
- The average rate of change from 4 to 9 years is 45/5 = 9 ft/yr.
- The average rate of change from 9 to 16 years is 63/7 ≈ 9 ft/yr.
Since the average rate of change is consistent at 9 ft/yr, the rate of change of the average height in feet of the trees on the farm with respect to the number of years since the trees were planted is 9 ft/yr. Therefore, the correct answer is 9 ft/yr.