Problem 83 - Precalculus with Limits

Question

The number \(N\) of trees of a given species per acre is approximated by the model \(N=68\left(10^{-0.04 x}\right)\) \(5 \leq x \leq 40,\) where \(x\) is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when \(N=21\).

Step-by-Step Solution

Answer

\(x = \frac{\ln\left(\frac{21}{68}\right)}{-0.04 \ln(10)}\) is the solution for \(x\), which is the average diameter of the trees in inches.

Step 1: Set the model equation equal to the given number of trees

First, replace \(N\) in the model equation with the given \(N=21\), which results to: \(21 = 68\left(10^{-0.04 x}\right)\)

Step 2: Solve for \(x\)

To find the value of \(x\) that satisfies the equation, you need to isolate \(x\). Begin by dividing both sides by 68, you get \(\frac{21}{68} = 10^{-0.04 x}\). Taking the natural logarithm (ln) on both sides will help isolate \(x\). Remember that \(\ln(10^a) = a \ln(10)\). So, you will get \(\ln\left(\frac{21}{68}\right) = -0.04x \ln(10)\), then divide both sides by \(-0.04 \ln(10)\), you finally obtain \(x = \frac{\ln\left(\frac{21}{68}\right)}{-0.04 \ln(10)}\)

Step 3: Calculate the diameter

Substitute the natural logarithm values into the obtained equation in step 2 and simplify to compute the diameter \(x\).

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms with an unknown variable. They are essential tools in solving equations where an exponent needs to be isolated, such as in exponential growth or decay problems.
The process usually involves using log identities to break down and simplify the equation so the variable can be isolated.
When solving logarithmic equations:

• Always ensure that the base of the logarithm and the argument remain positive.
• Use properties of logarithms, like the fact that \(\ln(a^b) = b \ln(a)\), to simplify the equation.
• Once simplified, if not apparent, convert the logarithmic equation to an exponential form.

With the original exercise, we have transformed the exponential equation by taking natural logs to an easier format for solving \(x\). This manipulation takes advantage of the inverse relationship between exponents and logarithms.

Diameter Approximation

Diameter approximation involves estimating the size of a circular object using given mathematical models. It's particularly useful in fields such as forestry, where measuring each tree's diameter individually isn't always feasible.
In forestry, the average diameter of trees in a plot is crucial for managing resources and understanding the ecosystem's health. These measurements can help predict tree growth patterns and potential yield.

• Diameter is often measured at a standard height, typically 3 feet from the ground in forestry surveys.
• Mathematical models like \(N = 68(10^{-0.04x})\) provide estimates to efficiently assess large areas.
• When applying such formulas, accuracy is dependent on the model's assumptions of the species and terrain.

In this exercise, we approximated the diameter of trees by rearranging the mathematical model to solve for \(x\), based on the expected number of trees.

Mathematical Modeling

Mathematical modeling is the process of creating abstract representations of real-world phenomena, allowing for predictions and problem-solving. It harnesses the power of mathematics to simulate scenarios which are otherwise difficult to measure directly.
Models can be represented by different mathematical forms, such as functions, equations, graphs, or statistics.

• Simple models capture the essence of the problem with minimal data, like the exponential model used in tree density.
• Models are tested and validated against real-world observations to ensure reliability.
• Once validated, they can be used to make predictions or to conduct 'what-if' scenarios for planning and decision-making.

In our exercise, the model \( N=68(10^{-0.04x}) \) serves as a tool to estimate the average diameter of trees. This relationship provides insights into the population density of trees with respect to their size.