Laura planted trees as a hobby at a tree farm. She records the average height in feet of trees and the number of years since the trees were planted. She noticed after 3 years the average height was 24 feet. She also recorded after 11 years the average height was 80 feet. 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?

To find the rate of change of the average height of the trees with respect to the number of years since they were planted, we can treat this situation as a linear relationship between time and height. The rate of change (slope) is given by the formula for the slope of a line:

\[
\text{slope} = \frac{\text{change in height}}{\text{change in time}} = \frac{h_2 - h_1}{t_2 - t_1}
\]

Where:
- \( h_1 = 24 \) feet (height after 3 years)
- \( h_2 = 80 \) feet (height after 11 years)
- \( t_1 = 3 \) years
- \( t_2 = 11 \) years

Now, plug these values into the formula:

\[
\text{slope} = \frac{80 - 24}{11 - 3} = \frac{56}{8} = 7 \, \text{feet per year}
\]

Thus, the rate of change of the average height of the trees is **7 feet per year**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How can you use this rate of change to predict the height of the trees after 15 years?
2. What would be the height of the trees after 0 years, assuming this rate of change holds?
3. How do you interpret a negative rate of change in such a scenario?
4. Can we model this situation with a linear equation? If so, what would the equation be?
5. How would the rate of change differ if the growth pattern was exponential rather than linear?

**Tip:** When dealing with word problems, always identify variables and their relationships clearly before attempting to solve the problem.

This problem explores how to calculate the rate of change of tree height over time based on the given data. Learn how to apply the slope formula to find the growth rate, with a detailed breakdown of the linear relationship between height and years. Suitable for Grades 6-8.

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