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We are given the ages and lengths (heights) of trees and are asked to find an exponential model for tree height as a function of age. We then need to use this model to predict the height for a 34-year-old tree.

### Exponential Model Equation
The general form of an exponential model is:

\[
L(t) = a \cdot e^{bt}
\]

Where:
- \(L(t)\) is the length (height) at age \(t\),
- \(a\) and \(b\) are constants to be determined from the data,
- \(e\) is the base of the natural logarithm.

### Given Data:
\[
\begin{array}{|c|c|}
\hline
\text{Age (years)} & \text{Length (feet)} \\
\hline
8 & 2.9 \\
11 & 6.2 \\
20 & 29.4 \\
\hline
\end{array}
\]

#### Step 1: Set up two equations from the data points to solve for \(a\) and \(b\).

Let's use the points (8, 2.9) and (20, 29.4) to create two equations:
1. \(2.9 = a \cdot e^{8b}\)
2. \(29.4 = a \cdot e^{20b}\)

#### Step 2: Solve these equations to find \(a\) and \(b\).

I'll now calculate the values of \(a\) and \(b\).The exponential model based on the given data is:

\[
L(t) = 0.619 \cdot e^{0.193t}
\]

Now, using this model, we can predict the height of a 34-year-old tree.

#### Step 3: Substitute \( t = 34 \) into the model to find the predicted height.The model predicts that the height of a 34-year-old tree would be approximately **438.49 feet**.

Would you like further details on the steps, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. How would the model change if the given tree heights were in meters instead of feet?
2. What are the implications of the exponential growth model in real-world tree height predictions?
3. How does the constant \(b\) affect the growth rate of the tree?
4. Can this model be used indefinitely, or are there limitations to applying exponential growth in biology?
5. How would you develop a logarithmic or linear model using the same data?

**Tip:** Exponential models are often used in biology to represent initial rapid growth that eventually slows down, but real-world factors can cause deviations from perfect exponential behavior.

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?

Learn how to develop an exponential model for tree height using given data points and predict the height of a 34-year-old tree. The problem involves using the formula L(t) = a * e^(bt) to determine the constants a and b from the data points (8, 2.9), (11, 6.2), and (20, 29.4). Suitable for students in Grades 9-12.

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