Problem 24 - College Algebra with Modeling and Visualization

Question

Near New Guinea there is a relationship between the number of bird species found on an island and the size of the island. The table lists the number of species of birds \(y\) found on an island with an area of \(x\) square kilometers. $$\begin{array}{rccccc}x\left(\mathrm{km}^{2}\right) & 0.1 & 1 & 10 & 100 & 1000 \\ \hline y \text { (species) } & 10 & 15 & 20 & 25 & 30\end{array}$$ (a) Find a function \(f\) that models the data. (b) Predict the number of bird species on an island of 5000 square kilometers. (c) Did your answer involve interpolation or extrapolation?

Step-by-Step Solution

Answer

(a) \(y = 5 \log_{10}(x) + 15\), (b) 34 species, (c) extrapolation.

Step 1: Analyze the Data

Examine the data points in the table, and recognize the relationship between the island area size, \(x\), and the number of bird species, \(y\). We're looking for a consistent pattern to establish a mathematical model.

Step 2: Determine the Type of Model

Notice that as \(x\) scales by a factor of 10, \(y\) increases by 5, suggesting a logarithmic relationship given the simplicity of the data. This resembles a logarithmic function form, because \(y\) doesn't increase linearly or exponentially with \(x\).

Step 3: Propose a Mathematical Model

Consider the function form \(y = a \, \log_{10}(x) + b\) to model the relationship. This assumes a base-10 log function due to how \(x\) grows uniformly by powers of 10.

Step 4: Calculate the Parameters of the Model

Using known values, - for \(x = 0.1\), \(y = 10\)- for \(x = 1\), \(y = 15\)solve the equations for \(a\) and \(b\): Substituting into the equation:1. \(10 = a \, \log_{10}(0.1) + b\) 2. \(15 = a \, \log_{10}(1) + b\)Solve these to find the values of \(a\) and \(b\). For \(\log_{10}(0.1) = -1\) and \(\log_{10}(1) = 0\), - From (1): \(10 = -a + b\) - From (2): \(15 = b\)Substitute \(b\) to get \(a\); \(-a + 15 = 10\), thus \(a = 5\). The model is \(y = 5 \, \log_{10}(x) + 15\).

Step 5: Use the Model to Predict

Predict the number of birds for \(x = 5000\). Substitute into the equation:\[y = 5 \, \log_{10}(5000) + 15\]Calculate \(\log_{10}(5000)\) which is approximately 3.699, thus \(y \approx 5 \times 3.699 + 15 \approx 33.495\).Therefore, approximately 34 species are predicted.

Step 6: Determine Interpolation or Extrapolation

Since 5000 sq km is beyond the range of the x-values in the table, predicting for 5000 sq km requires extrapolation, not interpolation.

Key Concepts

Logarithmic Function

In mathematical modeling, logarithmic functions are quite useful for depicting relationships where the rate of growth diminishes over time. In the given exercise, the relationship between the island area, denoted by \(x\), and the number of bird species, \(y\), appears to follow such a pattern.

The data suggests a logarithmic nature because as the area \(x\) increases by a factor of 10, the number of species \(y\) increases only incrementally, by 5. What's fascinating about logarithmic functions is how they handle such cases where the variable \(x\) scales dramatically, yet \(y\) experiences relatively small changes.

In our model, we use the function \(y = 5 \log_{10}(x) + 15\). Here, the logarithm base 10 is particularly suitable as our \(x\) values increase in powers of ten. The parameters \(a = 5\) and \(b = 15\) were calculated from the given data, representing the incremental increase in species per 10-fold area increase and the starting number of species at 1 km², respectively.

Interpolation and Extrapolation

Understanding interpolation and extrapolation is a crucial part of making predictions in data analysis. Interpolation involves estimating a value within the range of the known data points, whereas extrapolation involves predicting a value beyond this range.

In the exercise, predicting the number of bird species for an island with an area of 5000 square kilometers involves extrapolation. This is because the actual data set provides values between 0.1 and 1000 square kilometers.

Extrapolation can be more risky and less accurate than interpolation because it extends the trend beyond the observed data. The assumption is that the established pattern continues in the same manner, but unforeseen factors might alter this relationship. So, while our model estimates approximately 34 species for 5000 km², it's good to keep in mind that the farther we go from the observed data, the more careful we should be about the reliability of our predictions.

Data Analysis

Data analysis is the backbone of identifying patterns and building mathematical models like the one in this exercise. The process involves closely examining the data, identifying trends, and establishing a mathematical relationship that accurately describes the behavior of the data.

To begin with, analyzing the initial data set reveals a steady incremental increase in the number of bird species with every tenfold increase in island size. This consistent pattern hints at a logarithmic relationship rather than a linear or exponential one.

Using such insights, we then derive a model that approximates the data: \(y = 5 \log_{10}(x) + 15\). This involves solving equations for model parameters, where we plug in known values and calculate constants. The model then serves as a tool for predicting outcomes under new conditions, such as islands with unmeasured square kilometer sizes.

Ultimately, good data analysis empowers us to create models that are not only mathematically sound but also practically applicable, allowing for informed predictions and decisions in real-world situations.