, Task 1 & Print 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. Part A Assume the ball passes through the points (3, 8), (5, 25?) and (6, 5). Use this data to set up a system of three equations and three unknowns (a, b, and c) that will allow you to find the equation of the parabola. Write the system in the space provided. B I Y x* X; 15px v A v & v IZ = = E v 8- Space used (includes formatting): 0/ 15000 Part 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. Show your work and final equation in the space provided. BIU X X 2 15px A E E Space used (includes formatting): 0 / 15000Part 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. BIU X X 2 15px V A E E V Space used (includes formatting): 0 / 15000, Task 2 &1 Print 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. Your aerobics session should be at least 30 minutes. How many minutes should you perform each exercise to burn the maximum calories? Part A Write a system of inequalities and/or equations based on the given constraints. Then, note the objective function. B I U x* X, 12pt v A v v = = = M v B~ Space used (includes formatting): 0/ 15000 Part B Convert the inequalities into equations, and then use the substitution method to find three possible vertices in the form (x, y, z). BIUXX 2 12pt V A WN - E E Space used (includes formatting): 0 / 15000Part C Test the three 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. Show your work and final answer in the space provided. v il I i il B I U x* X, 15px v A v 2 v iz o= B v 8B Space used (includes formatting): 0 / 15000 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. Graphing Calculator Tip ( Now that you've learned how to use a graphing calculator to solve linear programming problems, solve the following problems. Part A 2r +y 0 Graph the inequalities on your graphing calculator, and find the vertex points of this system. v Inl " M il B I Y x* X, 12pt v A v S v == B v. B~ I Space used (includes formatting): 0 / 15000 Part B In part A, you obtained the vertex points of the given system. Test these vertex points in the objective function, f(z) = 4x + by using a graphing calculator. Find the maximum point. B IU X X 2 15px V A E E VB v Space used (includes formatting): 0 / 15000