Problem 103 - Applied Calculus

Question

\(103-104\). ENVIRONMENTAL SCIENCE: Biodiversity It is well known that larger land areas can support larger numbers of species. According to one study, multiplying the land area by a factor of \(x\) multiplies the number of species by a factor of \(x^{0.239}\). Use a graphing calculator to graph \(y=x^{0.239}\). Use the window [0,100] by [0,4]. Find the multiple \(x\) for the land area that leads to double the number of species. That is, find the value of \(x\) such that \(x^{0239}=2\). [Hint: Either use TRACE or find where \(y_{1}=x^{0.239}\) INTERSECTs \(y_{2}=2 .\).

Step-by-Step Solution

Answer

The required multiple x is approximately 5.93.

Step 1: Understand the Problem

We need to find the value of \( x \) such that \( x^{0.239} = 2 \). This implies we're looking for the \( x \)-value where the number of species is doubled.

Step 2: Setup the Equation

Our task is to solve the equation \( x^{0.239} = 2 \). To find this \( x \)-value, we can use a graphing calculator to intersect the functions \( y_1 = x^{0.239} \) and \( y_2 = 2 \).

Step 3: Graph the Functions

On a graphing calculator, input \( y_1 = x^{0.239} \) and \( y_2 = 2 \). Set the window size to \([0, 100]\) for \( x \) and \([0, 4]\) for \( y \) to ensure full visibility of the intersection point on the graph.

Step 4: Find the Intersection

Use the TRACE function or intersect feature on the calculator to find the point where \( y_1 \) intersects \( y_2 \). This intersection point gives the required \( x \)-value.

Step 5: Solve for x

After using the calculator, you will find the intersection point occurs at approximately \( x \approx 5.93 \). This means that multiplying the land area by approximately 5.93 will double the number of species.

Key Concepts

Biodiversity

Biodiversity refers to the variety of life found on Earth. This diversity is essential because it boosts ecosystem productivity, where each species plays a significant role. Different organisms offer unique contributions to the environment, from the food chain to nutrient cycling. This rich variety helps ecosystems cope with changes and stressors, like climate shifts or human impacts.
For instance:

• Plants contribute to oxygen production and carbon dioxide absorption.
• Animals and insects aid in pollination and seed dispersal.
• Microorganisms play a role in decomposing matter and cycling nutrients.

Biodiversity is crucial for ecosystem health and services, directly affecting human life by providing resources such as food and medicine.

Species-Area Relationship

The species-area relationship is a fundamental ecological principle that describes how the number of species increases with increasing area. This concept posits that larger habitats typically contain more species due to increased habitat diversity and availability of resources.
In mathematical terms, this relationship is often expressed by the formula:\[S = cA^z\] where:

• S is the number of species,
• A is the area,
• c and z are constants for a particular region or type of habitat.

This principle is essential for conservation efforts because it highlights the importance of preserving large natural areas to maintain species diversity and ecosystem services. Understanding this relationship helps conservationists make more informed decisions about habitat protection.

Graphing Functions

Graphing functions is a vital mathematical skill used to visualize relationships between variables. By plotting functions on a graph, one can discern patterns and interpret the behavior of various equations. Most often, the x-axis represents the input values, while the y-axis shows the output values, giving a clear picture of how these variables interact.
For example, the graph of the function \( y = x^{0.239} \) shows a curve that increases as x increases. This visual representation helps to clarify how the number of species changes with area in a species-area relationship context. Graphing calculators are useful tools for accurately plotting these functions, offering features such as trace and intersect to analyze specific points on the graph effectively.

Exponential Equations

Exponential equations are equations in which a constant base is raised to a variable exponent. These equations often represent growth and decay situations, like population growth, radioactive decay, or, in this case, the relationship between species number and land area size. An exponential equation generally has the form \( y = a \, b^x \), where \( a \) is a constant factor and \( b \) is the base that affects the rate of change.
Solving these equations requires specific techniques, such as using logarithms or graphing. In the species-area problem, the equation \( x^{0.239} = 2 \) signifies that we are looking for an x that would result in a doubling of the species number. This involves understanding how to manipulate and graph such equations to find the required x-values, illustrating the exponential nature of species growth with area expansion.