Math Problem Statement

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?

Solution

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⋅ebtL(t) = a \cdot e^{bt}

Where:

  • L(t)L(t) is the length (height) at age tt,
  • aa and bb are constants to be determined from the data,
  • ee is the base of the natural logarithm.

Given Data:

\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.

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Math Problem Analysis

Mathematical Concepts

Exponential Modeling
Algebra
Data Analysis

Formulas

L(t) = a * e^(bt)

Theorems

Exponential Growth Model

Suitable Grade Level

Grades 9-12

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