Problem 110 - College Algebra and Calculus: An Applied Approach

Question

The earnings per share \(y\) (in dollars) for Paychex, Inc. from 1996 to 2005 can be modeled by \(y=-0.0014 t^{2}+0.123 t-0.57, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Paychex, Inc.) (a) Sketch a graph of the equation. (b) In 2005 , Paychex predicted that its earnings per share would be \(\$ 1.22\) in 2006 and \(\$ 1.40\) in 2007 . Use the model to predict the earnings per share for these years. How well does the model support Paychex's predictions? (c) Paychex also predicted its earnings per share to reach $$\$ 1.90$$ sometime in 2009,2010, or \(2011 .\) How well does the model support Paychex's prediction?

Step-by-Step Solution

Answer

Conclusion depends on the results of the steps in the detailed solution. After calculating and comparing the earnings per share using the given equation and the company's prediction, an evaluation of the model's reliability and accuracy can be provided. The graphical representation will visiualize this.

Step 1: Calculation of earnings per share for 2006

Firstly, substitute \(t=16\) (representing 2006 since \(t=6\) stands for 1996) into the equation for y: \(y=-0.0014 * 16^{2}+0.123 * 16-0.57\)

Step 2: Calculation of earnings per share for 2007

Similar to step 1, substitute \(t=17\) into the equation for y: \(y=-0.0014* 17^{2}+0.123 * 17-0.57\)

Step 3: Comparison of model prediction with actual prediction for 2006 and 2007

Compare the calculated earnings per share for 2006 and 2007 with the company's prediction. This will offer insight on how well the model supports the company's predictions.

Step 4: Calculation of earnings per share for 2009, 2010 and 2011

Substitute \(t=19\) into the model to determine earnings per share for 2009. Repeat this step for \(t=20\) and \(t=21\) to predict earnings for 2010 and 2011 respectively.

Step 5: Comparison of model prediction with actual prediction for 2009 to 2011

Compare the calculated earnings per share for 2009, 2010, and 2011 with Paychex's prediction of \$ 1.90. This allows assessment of the model's consistency with these predictions.

Step 6: Graphical Representation

Plotting the equation \(y=-0.0014 t^{2}+0.123t-0.57\) for \(6 \leq t \leq 21\) will give a graphical representation of the earnings. The x-axis will denote the years from 1996 to 2011 (represented as 6 to 21) and the y-axis will denote the earnings per share in dollars.

Key Concepts

Graphing Quadratic Equations

Graphing quadratic equations can seem daunting at first, but it's a fairly straightforward process once you understand the basics. A quadratic equation typically takes the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
In our exercise, the equation given is \(y = -0.0014t^2 + 0.123t - 0.57\). Here, \(t\) represents the year since 1996, so \(t = 6\) corresponds to the year 1996.
To graph a quadratic equation:

• Identify the vertex. Since the parabola can open upwards or downwards, the vertex can indicate a maximum or minimum point on the graph. For our given equation, since the leading coefficient (\(-0.0014\)) is negative, it opens downwards.
• Determine the points where the curve intersects with the y-axis and the x-axis.
• In this context, the y-axis intersection represents the earnings when \(t\) equals zero (theoretically, the base rate of earnings).
• The x-intercepts provide a view of when the earnings would hit zero, although not always practical depending on the context.
• Utilize other points, like \(t = 6\) and \(t = 15\), to plot the graph accurately within the given range.

Plotting this equation from \(t = 6\) to \(t = 15\) will provide a visual representation of the earnings per year.

Predictive Modeling

Predictive modeling involves using mathematical equations and models to forecast future outcomes based on historical data. The quadratic model in our example allows us to predict Paychex's earnings per share.
The process involves substituting different values of \(t\) to predict future earnings. For example, when predicting the earnings in 2006, we use \(t = 16\) because \(t = 6\) is 1996.
Using the model equation:

• For 2006, substituting \(t = 16\) into the equation results in \(y = -0.0014(16)^2 + 0.123(16) - 0.57\), which calculates the earnings per share for that year.
• Similarly, for 2007, using \(t = 17\) will offer the next year's predicted value.
• These calculated predictions are then compared against actual predictions provided by the company to analyze the model's effectiveness.

Predictive modeling adds value by providing structured insights and a quantifiable analysis of potential financial outcomes.

Polynomial Functions

Polynomial functions form the backbone of much mathematical modeling, frequently appearing in subjects like physics, engineering, and economics. A polynomial function is an expression consisting of variables and coefficients.
In the case of the earnings per share model for Paychex, the polynomial is quadratic because it includes \(t^2\), making it a second-degree polynomial.
Key features of polynomial functions include:

• They can represent various forms of real-world phenomena, like earnings trends, as depicted in this model.
• The degree of the polynomial (highest power of the variable) indicates the number of turns or changes in direction the graph can have.
• The coefficients in the polynomial determine the shape and position of the graph.

When using polynomial functions for modeling, it's important to recognize that while they can provide a close approximation, they sometimes may not capture every nuance of real-world variations, particularly outside of the input range for which they were designed.