Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Peregrine Falcons The numbers of nesting pairs \(y\) of peregrine falcons in Yellowstone National Park from 2001 to 2005 can be approximated by the linear model \(y=3.4 t+13, \quad 1 \leq t \leq 5\) where \(t\) represents the year, with \(t=1\) corresponding to 2001\. (Sounce: Yellowstone Bird Report 2005) (a) The total number of nesting pairs during this five-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 3.4 t+13 \\ y \geq 0 \\ t \geq 0.5 \\\ t \leq 5.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total number of nesting pairs.

Step-by-Step Solution

Answer

The total number of nesting pairs over the five-year period is approximately 116 pairs.

Step 1: Graph the System of Inequalities

The desired region is presented by the system of inequalities \[y \leq 3.4 t+13\], \[y \geq 0\], \[t \geq 0.5\], and \[t \leq 5.5\]. When graphed, these equations will provide a trapezoid. The line \(y = 3.4t+13\) is the top of the trapezoid (connecting the points at \(t = 1\) and \(t = 5\)). The \(x\)-axis functions as the bottom edge of the trapezoid. The region is bounded on the left by the line \(x = 0.5\) and on the right by the line \(x = 5.5\).

Step 2: Calculate the Area of the Trapezoid

The formula for the area of a trapezoid is \[A = \frac{1}{2}h(b1+b2)\] where \(h\) represents the height of the trapezoid, and \(b1\) and \(b2\) represent the bases of the trapezoid. Here, the height of the trapezoid is \(h = 5.5 - 0.5 = 5\), while the bases are represented by the value of the function \(y = 3.4t + 13\) at \(t = 1\) and \(t = 5\), which are \(b1 = 3.4*1+13 = 16.4\) and \(b2 = 3.4*5+13 = 30\). Therefore, by substituting these values into the formula, the area can be calculated as \[A = \frac{1}{2} * 5 * (16.4 + 30) = 116\].

Step 3: Interpreting the Result

This calculation, which yielded the area of the trapezoid, gives an approximation of the total number of nesting pairs in the five-year period from 2001 to 2005.

Key Concepts

System of Inequalities

In algebra, a system of inequalities is a set of multiple inequalities that together define a region of possible solutions. When graphing a system of inequalities such as \[y \leq 3.4 t+13, \quad y \geq 0, \quad t \geq 0.5, \quad t \leq 5.5\],we are looking for the area where all these conditions are satisfied simultaneously.

To visualize this, you would draw the line \(y = 3.4t + 13\) and shade the area below it as this represents \(y \leq 3.4t + 13\). Next, you shade above the \(x\)-axis to satisfy \(y \geq 0\). The vertical lines \(t = 0.5\) and \(t = 5.5\) act as the left and right boundaries, restricting the region further. The result is a shaded trapezoidal area that represents all possible solutions to the system.

Area of a Trapezoid

The area of a trapezoid can be calculated using the formula \[A = \frac{1}{2}h(b1+b2)\], where \(h\) is the height and \(b1\) and \(b2\) are the lengths of the two parallel sides, often referred to as bases.

For example, in the provided system of inequalities, the bases of the trapezoid are determined by evaluating the function \(y = 3.4t + 13\) at the boundary values of \(t\), giving us \(b1 = 16.4\) and \(b2 = 30\). The height is the distance between these boundaries, which is \(5.5 - 0.5 = 5\). Plugging these into the formula, we get an area of 116, representing the total number of nesting pairs of peregrine falcons during the observed period.

Graphing Linear Equations

Graphing linear equations is a fundamental concept in algebra and is crucial for visualizing how equations behave. In a linear model such as \(y=3.4 t+13\), where \(t\) and \(y\) are variables representing the year and the number of nesting pairs respectively, the graph is a straight line.

Step-by-Step Graphing

To graph the equation, plot two points that lie on the line—usually by choosing two values for \(t\) and calculating the corresponding \(y\) values—and connect them with a straight line. This line forms one edge of our trapezoid and helps us understand the relationship between time and nesting pairs in graphical terms.

Applied Mathematics

The study of applied mathematics involves using mathematical methods and theories to solve real-world problems, like predicting population numbers in ecology. When we approximate the total number of nesting pairs of peregrine falcons by finding the area of the trapezoid, we are applying the principles of algebra and geometry to a practical problem in environmental science.

In applied mathematics, it is common to use models, such as the linear model \(y=3.4 t+13\), to represent complex scenarios in a simplified way. By graphing the linear model and corresponding inequalities, and calculating the area of shapes like trapezoids, we can gain valuable insights into trends and make predictions based on mathematical reasoning.