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# Unit Activity Unit: Mathematical PracticesYou will make sense of problems and solve them, use mathematics to model real-world situations, and use...

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© 2014 EDMENTUM, INC. Unit Activity Unit: Matrices This activity will help you meet these educational goals: Mathematical Practices—You will make sense of problems and solve them, use mathematics to model real-world situations, and use appropriate tools strategically. Introduction You have solved systems of equations and inequalities graphically, algebraically, and with matrices. You examined the algebra of matrices and studied the process of changing matrices to reduced echelon form. You learned to find the inverse and the determinant of a matrix, and to solve systems using Cramer’s rule. In this activity, you will apply these concepts to real-world situations. __________________________________________________________________________ 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 parabol that represents the ball’s path through the air. a. Assume the ball passes through the points . 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. Type your response here: 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. Type your response here: 1
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. Type your response here: 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. Type your response here: b. Convert the inequalities into equations, and then use the substitution method to find four possible vertices in the form ( x , y , z ). Type your response here: 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. Type your response here: 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 2 x + y ≤ 13, its slope-intercept form will be y ≤ 13 − 2 x . 2
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Unit Activity Unit: Matrices
Mathematical Practices—You will make sense of problems and solve them, use mathematics
to model real-world...

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