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.