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To solve this problem, we need to develop an exponential model for tree height as a function of tree age. The general form of an exponential model is:

\[
y = ab^x
\]
Where:
- \( y \) is the height (in feet),
- \( x \) is the age (in years),
- \( a \) is the initial height when \( x = 0 \),
- \( b \) is the growth factor.

We are given the following data points:
- \( (6, 2.2) \)
- \( (12, 5.3) \)
- \( (18, 26.4) \)

We will use these points to determine the values of \( a \) and \( b \), then use the model to predict the height of a 34-year-old tree.

I'll now calculate the parameters \( a \) and \( b \).The exponential model for tree height based on the given data is:

\[
y = 0.269 \cdot (1.290)^x
\]

Where \( y \) is the height in feet and \( x \) is the age in years. 

For a 34-year-old tree, the predicted height is approximately:

\[
1551.70 \text{ feet}
\]

Let me know if you would like more details or have any questions.

Here are 5 related questions for deeper understanding:
1. How does the exponential growth model differ from a linear growth model?
2. How can we verify if an exponential model is the best fit for this data?
3. What would be the height of a 20-year-old tree using this model?
4. How can we adjust the model to include environmental factors affecting growth?
5. What does the parameter \( b \) signify in this context?

**Tip:** When interpreting exponential models, the base \( b \) indicates the rate of growth, while the initial height \( a \) helps define the starting point.

The table below shows the ages of some trees and their corresponding heights. Find an exponential model for tree height as a function of age. What height does your model predict for a 34-year-old tree? Age (years): 6, 12, 18. Length (feet): 2.2, 5.3, 26.4.

Learn how to create an exponential model to predict tree height based on age. The problem involves finding a model from given data points (age 6, 12, 18) and predicting the height of a 34-year-old tree. Ideal for high school students studying exponential growth and algebra.

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