Problem 71 - 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 1.600 feet.

Step 1: Identify given values

We know the age of the tree, which is \( t = 20 \) years.

Step 2: Substitute age into the formula

Use the given logistic growth model for diameter: \[ D(t) = \frac{5.4}{1 + 2.9 e^{-0.01 t}} \]Substitute the given age \( t = 20 \)\[ D(20) = \frac{5.4}{1 + 2.9 e^{-0.01 \times 20}} \]

Step 3: Calculate the exponent

Calculate the value of the exponent:\(-0.01 \times 20 = -0.2\).This gives us the expression \( e^{-0.2} \).

Step 4: Evaluate the exponential function

Calculate \( e^{-0.2} \) using a calculator. This value is approximately 0.8187.

Step 5: Substitute and simplify

Substitute \( e^{-0.2} \approx 0.8187 \) into the denominator:\[ D(20) = \frac{5.4}{1 + 2.9 imes 0.8187} \]

Step 6: Perform arithmetic operations in the denominator

Multiply and add in the denominator:\( 2.9 \times 0.8187 \approx 2.37423 \), \( 1 + 2.37423 \approx 3.37423 \).

Step 7: Final calculation for diameter

Use the values calculated in previous steps:\[ D(20) = \frac{5.4}{3.37423} \approx 1.600 \]

Step 8: Interpret the result

The diameter of the 20-year-old tree is approximately 1.600 feet.

Key Concepts

Tree Diameter Calculation

When calculating the diameter of a tree using a logistic growth model, we take into account how the tree grows over time. The logistic growth model is a common way to model natural systems where growth starts quickly and then slows as it approaches a maximum value. For the given problem, the diameter of a tree at age 20 years can be predicted.
We start by identifying the specific formula which involves the tree's age and constants specific to the tree species.

• Identify the constants involved: In the formula, constants such as 5.4 and 2.9 represent species-specific growth limits and rates.
• Substitute the known age of the tree into the formula to calculate its diameter.

The subtraction and addition steps are crucial as they affect how the equation adjusts the diameter in response to the changing rate of growth over time.

Exponential Function Evaluation

The exponential function is a key part of the logistic growth model. It accounts for how certain processes amplifies or decays at shifting rates. When evaluating an exponential function such as \( e^{-kt} \), it helps to understand each part:

• In our formula, \( -kt \) serves as the exponent, where \( k \) is a given constant and \( t \) represents time.
• Multiply to find the precise value of the exponent, \( -0.01 \times 20 = -0.2 \).

Using a calculator, we solve \( e^{-0.2} \), which turns out to be roughly 0.8187.
This value decreases the growth rate effect because as \( t \) increases, the negative exponent slows down growth from the exponential function's rapid starting expansion.

Arithmetic Operations in Mathematics

Performing arithmetic operations accurately is fundamental when dealing with mathematical expressions, particularly in complex models like the logistic growth model. After evaluating the exponential component, it gets substituted back into the main formula.
The remaining arithmetic calculations involve:

• Multiplying values: The multiplication \( 2.9 \times 0.8187 \) gives approximately 2.37423.
• Adding in the denominator: Combine the calculated product with 1 to get 3.37423.
• Finally, divide the constant 5.4 by this sum to determine the diameter, \( \frac{5.4}{3.37423} \approx 1.600 \).

Each step of arithmetic involves basic operations but they combine to give a precise prediction of the tree's diameter. Keeping track of the sequence ensures that calculations are both accurate and efficient.