Problems 1. A calculator company produces a scientific calculator and a graphing calculator. Long- term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping con- tract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator pro- duces a $5 profit, how many of each type should be made daily to maximize net profits? (Hint: This is useful for your constraint equation. (a) Write out the inequalities and the constraint equation needed to solve this problem. You will need to define your own variables (i.e. x is scientific calculators.) (b) What do each of the constraint equations mean? (c) Graph the inequalities and define the feasible region. (d) What does this feasible region mean? (e) What are the corner points for this problem? What doe they mean? (f) What combination number (corner point) of scientific and graphing calculators maximizes the profit? (g) What combination number (corner point) of scientific and graphing calculators minimizes the profit? (h) What doe the solutions mean? Give a short explanation as to why these solutions make sense. EXTRA CREDIT (5pts): Suppose the profit, P, is now subject to P = 4x-7y, meaning scientific calculators generate $4 in profit and graphing calculators generate -$7 in profit (a loss), how does that change the maximum and minimum for the given corner points? Doe these solutions make sense? 2. Solve the following system of equations: -7x-6y-12z = -33 5x+5y+7z = 24 x+4z = 5 (a) Using Row-Echelon form (and back substitution) (b) Using reduced Row-Echelon form. (c) Find the inverse of A (A is the matrix you get from the system of equations without the solutions on the right hand side of the equal sign) (d) Show that A A-1 A-1 A-I, I is the identity matrix. = . EXTRA CREDIT (5pts): Why does this system have an inverse or as we would say, invertible? Explain your answer in a few sentences. 3. It's a race! You can run 0.2km (1/5km) every minute. The Horse can run 0.5km (1/2km) every minute. But it takes 6 minutes to saddle the horse. This is a very simple problem, you just have to think carefully about how you organize it. (a) Set up the equations needed to solve this system. Hint: Distance can be thought of as y, while time can be thought of as x. Put your equations in Standard Form. (b) Solve the system of equations by graphing. (c) Solve the system of equations by elimination. (d) Solve the system of equations by substitution. (e) How far can you get before the horse catches you? EXTRA CREDIT (5pts): Is this system inconsistent or consistent (explain why)? Is this system dependent or independent (explain why)? Give an example of a system that would be (a) inconsistent, (b) consistent-dependent, (c) consistent-independent.