Problem 25 - Applied Calculus

Question

The number of species of coastal dune plants in Australia \(2 E\) decreases as the latitude, in \(^{\circ} \mathrm{S}\), increases. There are 34 species at \(11^{\circ} \mathrm{S}\) and 26 species at \(44^{\circ} \mathrm{S} .^{19}\) \(3 i\) (a) Find a formula for the number, \(N\), of species of coastal dune plants in Australia as a linear function of the latitude, \(l\), in \(^{\circ}\) S. (b) Give units for and interpret the slope and the vertical intercept of this function. (c) Graph this function between \(l=11^{\circ} \mathrm{S}\) and \(l=\) \(44^{\circ} \mathrm{S} .\) (Australia lies entirely within these latitudes.)

Step-by-Step Solution

Answer

The linear function is \(N = \frac{-8}{33}l + 36.67\). The slope decreases by 0.242 species per degree increase in latitude.

Step 1: Identify Given Points

The problem provides two points that can be used to find a linear formula: at \(11^\circ \mathrm{S}\) there are 34 species, and at \(44^\circ \mathrm{S}\) there are 26 species. These can be expressed as the points \((11, 34)\) and \((44, 26)\) respectively.

Step 2: Calculate the Slope

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute \((x_1, y_1) = (11, 34)\) and \((x_2, y_2) = (44, 26)\) into the formula to find the slope.\[m = \frac{26 - 34}{44 - 11} = \frac{-8}{33}\]

Step 3: Find the Equation of the Line

Now that we have the slope, use the point-slope form of the equation \(y - y_1 = m(x - x_1)\) to find the linear equation. Plug in \(m = \frac{-8}{33}\), and use one of the points, say \((11, 34)\):\[N - 34 = \frac{-8}{33}(l - 11)\]\[N = \frac{-8}{33}l + \frac{8}{33} \times 11 + 34\]\[N = \frac{-8}{33}l + \frac{88}{33} + 34\]\[N = \frac{-8}{33}l + 2.67 + 34\]\[N = \frac{-8}{33}l + 36.67\]

Step 4: Interpret the Slope and Intercept

The slope of \(-\frac{8}{33}\) indicates that the number of species decreases by approximately 0.242 species per degree increase in latitude. The vertical intercept 36.67 suggests that extrapolating back to \(0^\circ\) S, the model predicts about 36.67 species.

Step 5: Graph the Function

The graph of the function \(N = \frac{-8}{33}l + 36.67\) is a straight line starting from the point \((11, 34)\) to \((44, 26)\). Plot these points on the graph, and draw the line connecting them to visually represent the decrease in species with increasing latitude.

Key Concepts

Slope Interpretation

Understanding the slope of a line is essential when working with linear functions, as it provides crucial information about the rate of change from one point to another. In this exercise, we calculate the slope using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, the variables \(x_1, x_2\) are the latitudes—and \(y_1, y_2\) are the number of plant species at those latitudes.
The calculated slope value is \(m = -\frac{8}{33}\), which represents the rate of decrease in the number of plant species as the latitude increases.
In practical terms, a slope of \(-\frac{8}{33}\) means that the number of coastal dune plant species decreases by approximately 0.242 species per one-degree increase in latitude.
This negative slope confirms the trend that species numbers decline as we move southward in Australia, highlighting the importance of latitude on biodiversity.

Graphing Linear Functions

Graphing a linear function involves plotting the line on a coordinate plane based on the linear equation derived from the problem. The linear function for this exercise is:

• \( N = -\frac{8}{33}l + 36.67 \)

This equation helps us understand how the number of species changes over different latitudes.
To graph this function, identify and plot the points given in the problem, namely \((11, 34)\) and \((44, 26)\), on a coordinate plane.
Draw a line through these points, ensuring it extends from the starting latitude \(11^\circ S\) to the ending latitude \(44^\circ S\).
The resulting graph is a straight line that descends from left to right, visually indicating the decreasing number of plant species as the latitude increases.
This graphical representation gives us a clear visual understanding of the rate and extent of change in species numbers across the latitudes considered.

Point-Slope Form

The point-slope form is a cornerstone of linear equations, providing a method to easily write an equation of a line when given a point and the slope. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

For this specific exercise, where \( m = -\frac{8}{33} \) and a point such as \( (11, 34) \) is given, substitute these values into the point-slope form to scaffold the line's equation.
This substitution leads to:

• \( N - 34 = -\frac{8}{33}(l - 11) \)

Solving this gives us the linear equation \( N = -\frac{8}{33}l + 36.67 \).
Using the point-slope form is crucial because it allows one to derive the linear relationship from known data, facilitating predictions and interpretations of linear trends.
This form effectively bridges the gap between abstract slope calculations and tangible real-world applications, like predicting species numbers at varying latitudes.