Problem 82 - A Graphical Approach to Precalculus with Limits

Question

A research biologist has shown that the number \(S\) of different plant species on a Galápagos Island is related to the area \(\mathscr{A}\) of the island by $$S=28.6 \sqrt[3]{\mathscr{A}}$$ Find \(S\) for an island with each area. (a) 100 square miles (b) 1500 square miles

Step-by-Step Solution

Answer

(a) 133 species, (b) 327 species.

Step 1: Understand the Formula

The given formula is used to find the number of different plant species, \(S\), on an island based on its area, \(\mathscr{A}\). The formula is \(S = 28.6 \sqrt[3]{\mathscr{A}}\). We will use this formula to calculate \(S\) for different values of \(\mathscr{A}\).

Step 2: Calculate Number of Species for 100 Square Miles

Substitute \(\mathscr{A} = 100\) into the formula. Calculate the cube root of \(100\) and then multiply the result by 28.6. \[ S = 28.6 \times \sqrt[3]{100} \]First, find \( \sqrt[3]{100} \approx 4.64 \). Then, multiply:\[ S = 28.6 \times 4.64 \approx 132.704 \]So, for an island with an area of 100 square miles, there are approximately 133 plant species.

Step 3: Calculate Number of Species for 1500 Square Miles

Substitute \(\mathscr{A} = 1500\) into the formula. Calculate the cube root of \(1500\) and then multiply the result by 28.6. \[ S = 28.6 \times \sqrt[3]{1500} \]First, find \( \sqrt[3]{1500} \approx 11.45 \). Then, multiply:\[ S = 28.6 \times 11.45 \approx 327.37 \]Thus, for an island with an area of 1500 square miles, there are approximately 327 plant species.

Key Concepts

Cube Root

Understanding cube roots is crucial when dealing with models like the one used to calculate plant species on an island. The cube root of a number is the value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3 because:

• 3 multiplied by 3 equals 9
• 9 multiplied by 3 equals 27

Cube roots are denoted as \( \sqrt[3]{x} \), where \( x \) is the number you want to find the cube root of. In our exercise, the biologist used the formula:

\[ S = 28.6 \sqrt[3]{\mathscr{A}} \]to show the relationship between the island area and the number of species. Understanding how to compute these roots, helps in solving real-world problems like estimating plant diversity based on land area. To solve this kind of equation:

• Find the cube root of the given area \( \mathscr{A} \)
• Multiply the cube root by the constant 28.6
• Interpret the result as the approximate number of species

This formula makes use of the mathematical property that as the area grows, the number of species grows at a rate dictated by the cube root.

Mathematical Model

Mathematical models are formulas or equations used to describe a real-world situation. They help scientists predict outcomes based on certain inputs. In this exercise, the mathematical model links the land area of the Galápagos Islands to the diversity of plant species.

Using the formula \( S = 28.6 \sqrt[3]{\mathscr{A}} \), we can calculate how many plant species are likely present based on the island's size. Here are key elements of a mathematical model:

• **Constants**: such as 28.6 in the formula, help to scale the relationship for specific conditions or datasets.
• **Variables**: like \( \mathscr{A} \), are the changing quantities that affect outcomes.
• **Predictions**: allow us to forecast results, for example, how adding more land would impact species diversity.

Mathematical models, when applied accurately, give us a powerful tool to make sense of complex data, illustrating relationships that may not be obvious otherwise. They provide insight into how changes in one part of a system can influence the whole.

Species-Area Relationship

The species-area relationship is an ecological concept describing how the number of species increases with the area surveyed. This relationship usually follows the principle that "more area equals more species," as seen in the exercise.

Applying this concept, if you double the area, it doesn't necessarily mean you will double the species count, due to diminishing returns as area grows larger. Instead, species diversity hits certain limits due to habitat requirements and available resources. The formula \( S = 28.6 \sqrt[3]{\mathscr{A}} \) approximates this relationship:

• The model suggests a robust but not linear correlation, showing gradual increase in species richness with increased area.
• Cube root transformation captures how species accumulation slows as more area is added.
• This makes the formula practical in conservation, helping prioritize which areas to protect based on potential biodiversity gains.

Understanding this relationship aids ecologists and conservationists to predict how species might cope with habitat fragmentation and changes, giving them the information needed to protect ecosystems effectively.