Problem 96 - Brief Applied Calculus

Question

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 triple the number of species. That is, find the value of \(x\) such that \(x^{0.239}=3\). [Hint: Either use TRACE or find where \(y_{1}=x^{0.239} \quad\) INTERSECTs \(\left.y_{2}=3 .\right]\)

Step-by-Step Solution

Answer

The value of \(x\) is approximately 165.8.

Step 1: Understand the Problem

We need to find the value of \(x\) such that \(x^{0.239} = 3\). This means we are looking for the multiple of the land area \(x\) that results in tripling the number of species.

Step 2: Graph the Function

Start by graphing the function \(y = x^{0.239}\) using a graphing calculator. Set your window to \(x\) values from 0 to 100 and \(y\) values from 0 to 4.

Step 3: Graph the Constant Function

Next, graph the horizontal line \(y = 3\) on the same graph. This represents the desired tripling of the number of species.

Step 4: Find the Intersection Point

Using the graph, find the intersection point of the two graphs \(y = x^{0.239}\) and \(y = 3\). This can be done using the TRACE feature or by calculating the intersection using the calculator's INTERSECT function.

Step 5: Solve for x

The \(x\)-value of the intersection point is the solution to \(x^{0.239} = 3\). Use the calculator to determine this value.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable power. These functions play a crucial role in modeling real-world phenomena where growth or decay occurs at a constantly proportional rate. In the exercise provided, the function is expressed as \(y = x^{0.239}\). Here, \(x\) is the base representing the factor by which the land area is multiplied, and \(0.239\) is the exponent, which determines the growth rate of the number of species.
The characteristic feature of exponential functions is their rate of change. Unlike linear functions, whose growth rate is constant, exponential functions grow or decrease at rates proportional to their current value. This nature makes them very useful for modeling situations like population growth, radioactive decay, and, as in this exercise, the species-area relationship. When graphed, exponential functions can start out very slowly but quickly rise or fall.

• The base \(x\) determines the type of exponential behavior: values of \(x > 1\) result in growth, \(0 < x < 1\) in decay.
• The exponent magnitude indicates how fast the growth will occur.

By understanding these properties, you can predict behavior and solve for variables, such as in our equation, where we need to find \(x\) that results in thrice the species.

Graphical Solution

Graphical solutions involve visually displaying equations or data to find intersections or trends rather than algebraic manipulation. In understanding the exercise, a graphical solution comes into play when we graph the function \(y = x^{0.239}\) using a graphing calculator. This method allows us to visually assess where this exponential curve intersects with another horizontal line representing a constant value, in this case, \(y = 3\).
The key steps in finding a graphical solution include: setting an appropriate window on the graphing tool, inputting the necessary functions, and identifying the point of intersection. In our task:

• The graphing window is set from \([0, 100]\) on the x-axis and \([0, 4]\) on the y-axis, to ensure visibility of relevant behavior.
• The TRACE feature or the INTERSECT function on the graphing calculator helps in locating the intersection point. These tools provide the x-value where the curve \(y = x^{0.239}\) crosses the line \(y = 3\).

This graphical method is particularly advantageous when dealing with complex exponential or non-linear equations, enabling a clearer understanding and practical solution without intensive calculations.

Species-Area Relationship

The species-area relationship is an ecological principle that describes how the number of species increases with the area surveyed. This relationship typically follows a pattern that can be expressed with exponential equations. According to the exercise, multiplying the land area by a factor of \(x\) leads to a change in the number of species by a factor of \(x^{0.239}\).
More simply, this relationship indicates that as you increase the size of the area you're looking at, the number of different species you observe will generally increase, often at a rate that's less than linear but follows a consistent mathematical rule. The species-area relationship can be influenced by several factors:

• Habitat diversity: Larger areas may contain more varied habitats, welcoming more species.
• Population size: More space often allows larger populations, though this isn't strictly linear.
• Isolation: Remote areas might show different characteristics due to limited species immigration.

Understanding this relationship is important for environmental conservation strategies and biodiversity studies, as it provides insight into how land-use change can impact species richness and distribution.