Problem 28 - Algebra and Trigonometry

Question

Tree Diameter For a certain type of tree the diameter \(D\) (in feet) depends on the tree's age \(t\) (in years) according to the logistic growth model $$ D(t)=\frac{5.4}{1+2.9 e^{-0.01 t}} $$ Find the diameter of a 20 -year-old tree.

Step-by-Step Solution

Answer

The diameter of a 20-year-old tree is approximately 1.60 feet.

Step 1: Understanding the Problem

We need to find the diameter of a tree that is 20 years old using the given logistic growth model. The model is presented as: \[ D(t) = \frac{5.4}{1 + 2.9 e^{-0.01 t}} \] where \(t\) is the age of the tree in years.

Step 2: Substituting the Age

Substitute \(t = 20\) into the equation to find the diameter of a 20-year-old tree.\[ D(20) = \frac{5.4}{1 + 2.9 e^{-0.01 \times 20}} \]

Step 3: Calculating the Exponential Term

First, calculate the exponential term in the denominator: \[ e^{-0.01 \times 20} = e^{-0.2} \]. Use a calculator to find \(e^{-0.2} \approx 0.8187\).

Step 4: Simplifying the Denominator

Now substitute \(e^{-0.2}\) into the denominator:\[ 1 + 2.9 \times 0.8187 \approx 1 + 2.37423 = 3.37423 \].

Step 5: Final Calculation

Now perform the final calculation to find \(D(20)\):\[ D(20) = \frac{5.4}{3.37423} \]. Use a calculator to find the result to get \(D(20) \approx 1.60 \) feet.

Key Concepts

Tree Diameter

Tree diameter is a crucial indicator of a tree's overall size and health. The diameter of a tree is typically measured at breast height, which is about 4.5 feet above the ground. In mathematical models, the diameter can also be predicted based on various factors such as the tree's age. In the given example, the tree's diameter increases over time following a logistic growth model. This model is particularly useful for capturing the way trees grow at different rates throughout their lifecycle.

• An initial rapid growth phase as young trees expand quickly.
• A slowing growth rate as trees mature and the diameter increases more gradually.
• A potential plateau as trees reach their maximum sustainable size.

Understanding the diameter is essential for foresters and environmental scientists when assessing forest health and predicting future forest conditions.

Age Function

The age function, in the context of this problem, provides a way to calculate the size of the tree based on its age. This approach allows us to understand the relationship between age and growth patterns. In this example, as the tree ages, the diameter grows according to a specific formula given by the logistic growth model.The function is written as:\[ D(t) = \frac{5.4}{1 + 2.9 e^{-0.01 t}} \]where:

• \(D(t)\) represents the diameter as a function of age \(t\).
• The constants in the formula determine the growth rate and the maximum diameter the tree can reach.
• As \(t\) increases, the effects of the exponential term decrease, depicting the slowing growth as maturity is reached.

This function helps in predicting how a tree will grow over the years and can be critical in planning conservation efforts and managing resources effectively.

Exponential Function

Exponential functions play an integral role in mathematical models of growth. These functions are characterized by their rapid change at one end and slower growth at the other, making them ideal for representing natural processes like tree growth. In our logistic growth model, the exponential function appears as part of the denominator:\[ e^{-0.01 t} \]

• The negative exponent \(-0.01t\) suggests a decay component, impacting how tree growth slows over time.
• As \(t\) increases, \(e^{-0.01t}\) becomes smaller, transitioning the growth curve from steep to more gradual.
• This behavior aligns well with the biological realities of tree growth, where rapid growth occurs at early stages but diminishes as the tree matures.

Using exponential functions to model growth helps in achieving predictions that mirror real-world scenarios. It is a powerful mathematical tool for understanding complex biological and environmental systems.