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!