Problem 119 - College Algebra with Modeling and Visualization

Question

The table lists numbers of Wal-Mart employees \(E\) in millions, \(x\) years after 1987 . $$ \begin{array}{cccccc} x & 0 & 5 & 10 & 15 & 20 \\ \hline B & 0.20 & 0.38 & 0.68 & 1.4 & 2.2 \end{array} $$ (a) Evaluate \(E(15)\) and interpret the result. (b) Find a quadratic function \(f\) that models these data. (c) Graph the data and function \(f\) in the same \(x y\) -plane. (d) Use \(f\) to estimate the year when the number of cmployees may reach 3 million.

Step-by-Step Solution

Answer

(a) 2002 had 1.4 million employees. (b) \(f(x) = 0.004x^2 + 0.008x + 0.20\). (d) Reach 3 million around 2012.

Step 1: Evaluate E(15)

Given the table, locate where \(x = 15\). The number of employees, \(E(15)\), corresponding to \(x = 15\) is given as \(1.4\) million. Therefore, \(E(15) = 1.4\). This means that 15 years after 1987, in 2002, Wal-Mart had 1.4 million employees.

Step 2: Determine the Interpolation Model

To find a quadratic function \(f(x) = ax^2 + bx + c\) that fits the data, we'll use the method of least squares or another fitting software as this step involves higher mathematics. Using the data points (0, 0.20), (10, 0.68), and (20, 2.2), the resultant quadratic function might be \(f(x) = 0.004x^2 + 0.008x + 0.20\). You can verify this using a graphing calculator or fitting software to match these data points accurately.

Step 3: Graph the Data and Function

Using graph plotting software or graph paper, plot the data points \((x, E)\) from the table: (0, 0.20), (5, 0.38), (10, 0.68), (15, 1.4), and (20, 2.2). Then, graph the quadratic function found in Step 2, \(f(x) = 0.004x^2 + 0.008x + 0.20\), on the same set of axes. The curve should pass close to or through the data points, showing the trend of employee growth over time.

Step 4: Estimate the Year for 3 Million Employees

Solve for \(x\) in the equation \(f(x) = 3\) with the quadratic function \(0.004x^2 + 0.008x + 0.20 = 3\). Rearrange to \(0.004x^2 + 0.008x - 2.8 = 0\). Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 0.004\), \(b = 0.008\), \(c = -2.8\), we calculate \(x\approx 24.5\). Hence, around 24.5 years after 1987, approximately in 2012, the employee count may reach 3 million.

Key Concepts

Interpolation Model

An interpolation model is a mathematical way to estimate values between known data points. It is particularly useful when you have a set of data points and want to predict intermediate values. In this exercise, we use a quadratic function for interpolation, which is expressed in the form \( f(x) = ax^2 + bx + c \). This type of function is ideal here because it can capture changes in growth not well represented by a simple linear model.
To create a quadratic interpolation model for our data:

• Begin by selecting the data points you want to include in your model. Here, points such as (0, 0.20), (10, 0.68), and (20, 2.20) are selected.
• Use tools like graphing calculators or software capable of handling quadratic regression. This software will calculate the coefficients \(a\), \(b\), and \(c\) to best fit the chosen points.

The goal is to determine the most accurate curve that represents the growth pattern of Wal-Mart's employee numbers over the years.

Least Squares Method

The least squares method is a mathematical approach used for fitting a curve to data points in a way that minimizes the sum of the squares of the differences between observed values and the values predicted by the model. When applied to quadratic functions, it helps find the best-fit coefficients \(a\), \(b\), and \(c\) of the equation \(f(x) = ax^2 + bx + c\).

This method involves several steps:

• Compute the predicted values \(f(x_i)\) for each data point \(x_i\) using the initial guessed coefficients.
• Calculate the difference \(f(x_i) - y_i\) for each data point \(y_i\), which represents the actual value.
• Square each difference, then sum all squared values for all data points.
• Adjust the coefficients \(a\), \(b\), and \(c\) to minimize this sum.

By following these steps, the least squares method provides a model that predicts data with the smallest possible deviations. This makes it a powerful tool in statistics and data analysis.

Graph Plotting

Graph plotting is the visual representation of data and mathematical models on a coordinate system, which helps us to understand patterns and relationships at a glance. In this exercise, plotting the data points given by Wal-Mart's employee numbers and the fitted quadratic function allows us to compare real growth with our estimated model visually.

To plot the graph:

• Start by setting up the axes: typically, the x-axis represents time in years since 1987, and the y-axis represents the number of employees in millions.
• Next, plot the individual data points provided by the exercise: (0, 0.20), (5, 0.38), (10, 0.68), (15, 1.4), and (20, 2.2).
• Now, draw the quadratic function \(f(x) = 0.004x^2 + 0.008x + 0.20\) you determined in the interpolation step. This will appear as a smooth curve passing close to the data points.

Graph plotting is crucial as it provides a clear visual verification of how well the quadratic model fits the actual data.

Quadratic Formula

The quadratic formula is a standard method for finding the solutions to a quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula calculates the roots of the equation, which are the x-values where the quadratic function intersects the x-axis.
In our exercise, once we have the quadratic function \(f(x) = 0.004x^2 + 0.008x + 0.20\), we use the formula to find when Wal-Mart's employee count may reach 3 million:

• Set \(f(x) = 3\), leading to the equation \(0.004x^2 + 0.008x - 2.8 = 0\).
• Plug \(a = 0.004\), \(b = 0.008\), and \(c = -2.8\) into the quadratic formula.
• Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
• Finally, solve for \(x\) using the formula. In this case, \(x \approx 24.5\), indicating that approximately 24.5 years after 1987, Wal-Mart's employee numbers might hit 3 million.

The quadratic formula ensures precise solutions to quadratic equations, making it indispensable for such interpolations.