Directions and Analysis

Task 1: Matrices and Linear Systems of Equations

Imagine that as a ball is tossed, its motion is tracked on a coordinate plane.  Given only a few of the points that the ball passes through, it is actually possible to determine the equation of the parabola that represents the ball's path through the air.

a. Assume the ball passes through the points (3, 8), (5, 20/3) and (6, 5). Use this data to set up a system of 3 equations and 3 unknowns (a, b, and c) that will allow you to find the equation of the parabola.

b. Use matrix manipulation to solve for a, b, and c. Set up a matrix equation for AX = B based on the system of equations you derived in part b where X is a matrix of the variables a, b, and c. Then, use Gauss-Jordan elimination to find the inverse of A. Finally, use your results to write the equation of the parabola.

c. Now that you have determined the equation of the parabola, assume that x represents the number of seconds that have passed since the ball was thrown and determine approximately how long it will take for the ball to hit the ground.

Task 2: Linear Programming

Imagine you are trying to maximize the calories you burn in a 60-minute workout you do a few times a week. Running burns 9 calories per minute, aerobics burns 6 calories per minute, and rowing burns 7 calories per minute. You want to perform all three exercises to work different muscle groups. For the best effect, you need to run for at least 5 minutes and row for at least 15 minutes. And your aerobics session should be no more than 30 minutes. How many minutes should you perform each exercise to burn the maximum calories?

a. Write a system of inequalities based on the given constraints. Then note the objective function.

b. Convert the inequalities into equations, and then use the substitution method to find four possible vertices in the form (x, y, z).

c. Test the four possible vertices that you found in the objective function in part a. Use those values to determine which set of values maximizes the objective function.

Task 3: Solving Linear Programming Problems Using a Calculator

Linear programming problems can also be solved using a graphing calculator. The following instructions will work for most commonly used graphing calculators, but note that they might vary slightly based on calculator model.

To use a graphing calculator to solve linear programming problems:

1. Convert the inequalities to the slope-intercept form; that is, write the inequalities in terms of y. For example, if the inequality is 2x + y ≤ 13, its slope-intercept form will be y ≤ 13 - 2x.

2. Enter the equation into the graphing calculator. Go to the Y= screen and enter the inequality as an equation. You can enter multiple inequalities (Y₁, Y₂, Y₃, and so on).

3. Set the windows for the graph. These values will depend upon your equations and the values you expect to see in the graph window.

4. Before you can graph the inequalities, you need to change the = sign to ≤ or ≥ (depending upon the inequality). You can do this by going to the Y= screen, moving the cursor to the left of the Y= symbol, and then pressing Enter. This will present the alternatives to the = sign: >, <, ≥, ≤. Select the inequality and then press Graph to see its graph with the shaded region (as per the inequality sign chosen).

5. You can also evaluate the objective function at the vertex points that you obtain after graphing the inequalities. Press Mode on the calculator. The second to last line will have three options: Full, Horiz, and G-T. By default, you're in the full-screen mode. You can change it to horizontal mode by selecting Horiz, which will split the screen into home screen options at the bottom and the graph on top. The G-T option splits the screen vertically, with the home screen options on the right and the graph on the left. You can move the cursor on the graph when you're in the Graph mode, and use the home screen options when you're in the Home screen mode. To toggle between these modes, you can press Graph (when you want to explore the graph) and 2nd Mode (when you want to explore the home screen options). Now when you want to evaluate the objective function at the vertex points, press Graph and then move the cursor to any of the vertex points. This will store the x and y values of that particular point on the graph. Then press 2nd Mode to go to the home screen, enter the objective function, and press Enter. This will evaluate the objective function at the points you selected on the graph in the Graph mode.

Now that you've learned how to use a graphing calculator to solve linear programming problems, solve the following problems

a. Consider the system 

Graph the inequalities on your graphing calculator, and find the vertex points of this system.

b. In part a, you obtained the vertex points of the given system. Test these vertex points in the objective function, f(z) = 4x + 6y using a graphing calculator. Find the maximum point.