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 30-year-old tree? 

 

Age (years)	8	13	18
Length (feet)	2.4	11.7	29.2
 

If you round your values when you write down the model, use at least three decimal places.


To find an exponential model for tree height as a function of age, we need to model the data in the form:

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

where:
- \( H(t) \) is the height of the tree as a function of age \( t \),
- \( a \) and \( b \) are constants that we need to determine,
- \( e \) is the base of the natural logarithm.

### Step 1: Set up equations based on given data points
We have three points from the table:
- Age \( t_1 = 8 \), Height \( H_1 = 2.4 \),
- Age \( t_2 = 13 \), Height \( H_2 = 11.7 \),
- Age \( t_3 = 18 \), Height \( H_3 = 29.2 \).

Using the model \( H(t) = a \cdot e^{bt} \), we get the following system of equations:

1. \( 2.4 = a \cdot e^{8b} \)
2. \( 11.7 = a \cdot e^{13b} \)
3. \( 29.2 = a \cdot e^{18b} \)

### Step 2: Solve the system of equations

#### 2.1: Divide Equation 2 by Equation 1 to eliminate \( a \):
\[
\frac{11.7}{2.4} = \frac{a \cdot e^{13b}}{a \cdot e^{8b}} \Rightarrow \frac{11.7}{2.4} = e^{5b}
\]
\[
4.875 = e^{5b}
\]
Take the natural logarithm of both sides:
\[
\ln(4.875) = 5b \Rightarrow b = \frac{\ln(4.875)}{5}
\]

#### 2.2: Calculate \( b \):
\[
b \approx \frac{1.584}{5} = 0.3168
\]

#### 2.3: Use Equation 1 to find \( a \):
Now substitute \( b \approx 0.3168 \) into Equation 1:
\[
2.4 = a \cdot e^{8(0.3168)}
\]
\[
2.4 = a \cdot e^{2.5344}
\]
\[
2.4 \approx a \cdot 12.614
\]
\[
a \approx \frac{2.4}{12.614} \approx 0.1903
\]

### Step 3: Write the exponential model

Now that we have \( a \approx 0.1903 \) and \( b \approx 0.3168 \), the exponential model is:
\[
H(t) = 0.190 \cdot e^{0.317t}
\]

### Step 4: Predict the height of a 30-year-old tree

Substitute \( t = 30 \) into the model:
\[
H(30) = 0.190 \cdot e^{0.317 \cdot 30}
\]
\[
H(30) = 0.190 \cdot e^{9.51}
\]
\[
H(30) \approx 0.190 \cdot 13461.883 \approx 2557.757 \text{ feet}
\]

So, the model predicts that a 30-year-old tree will be approximately **2557.757 feet** tall.

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Would you like further details on the model or solution steps? 

### Related Questions:
1. How would the model change if the tree growth rate decreased over time?
2. How does an exponential model compare to a linear model for this data?
3. What are other real-world phenomena that follow exponential growth?
4. How can we verify the accuracy of the model for ages not given in the table?
5. Can a logarithmic model provide a better fit for this data?

### Tip:
Exponential models are commonly used for data that grows at an increasing rate, but always check if other models (e.g., logarithmic, polynomial) fit better!

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 30-year-old tree? 

Age (years): 8, 13, 18
Length (feet): 2.4, 11.7, 29.2

Learn how to develop an exponential model for tree growth using data on tree ages and heights. This problem includes key steps such as solving for constants and predicting the height of a 30-year-old tree. Data points: Ages (8, 13, 18 years), Heights (2.4, 11.7, 29.2 feet). Suitable for Grades 10-12.

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