The Shirts Happen Clothing Company manufactures men's shirts and women's blouses. The production process includes cutting, sewing, and packaging. The table below gives the labor requirements (in minutes) for each type of garment: Minutes per garment Cutting Sewing Packaging Men's shirts 30 80 12 Women's blouses 60 60 4 The maximum number of labor hours available per day at Shirts Happen are 25 hours for cutting, 35 hours for sewing, and 5 hours for packaging. 1. Let x represent the number of men's shirts produced per day by the Shirts Happen Clothing Company, and let y represent the number of women's blouses produced per day. Create a system of five linear inequalities involving x and y that, when solved, give the set of all ordered pairs (x, y) of number of men's shirts and number of women's blouses that Shirts Happen can produce per day, given the constraints above. Next to each inequality, write a brief description or interpretation of its meaning. 2. Carefully graph the system of linear inequalities on a separate sheet of graph paper and then shade the feasible region. Is it bounded or unbounded? Identify and label on your graph the coordinates of each of the corner points. Attach the graph along with your work. 3. For each of the following solutions, determine whether or not it is a feasible solution to the Shirts Happen Clothing Company problem. Show your work and explain your reasoning. (a). 5 men's shirts per day, 20 women's blouses per day (b). 25 men's shirts per day, 25 women's blouses per day (c). 0 men's shirts per day, 0 women's blouses per day (d). ?10 men's shirts per day, ?20 women's blouses per day (e). 10 men's shirts per day, 10 women's blouses per day (f). 18 men's shirts per day, 14 women's blouses per day 4. Suppose that each men's shirt produced and sold yields a $25 profit and each women's blouse produced and sold yields a $50 profit. (a). An isoprofit line is a line of solution points (x, y) within the feasible region that each yield the same profit. Find two feasible solutions to the Shirts Happen Clothing Company problem that produce a total profit of $200, and then find two feasible solutions that produce a total profit of $500. Explain your answers, and then draw these two isoprofit lines on your graph ? one for a total profit of $200, and the other for a total profit of $500. (b). Finding optimal solution(s): How many men's shirts and women's blouses should the Shirts Happen Clothing Company produce each day so as to maximize its daily profit? What is the company's maximum daily profit as a result? Hint: There are actually multiple optimal solutions. List and explain two of them. Would there be a reason for the Shirts Happen Clothing Company to favor one optimal solution over another? Clearly explain your reasoning along with any additional assumptions that need to be made in determining an optimal solution to this problem.

4. Suppose that each men's shirt produced and sold yields a $25 profit and each women's blouse produced and sold yields a $50 profit. (a). An isoprofit line is a line of solution points (z, y) within the feasible region that each yield the same profit. Find two feasible solutions to the Shirts Happen Clothing Company problem that produce a total profit of $200, and then find two feasible solutions that produce a total profit of $500. Explain your answers, and then draw these two isoprofit lines on your graph - one for a total profit of $200, and the other for a total profit of $500. (b). Finding optimal solution(s): How many men's shirts and women's blouses should the Shirts Happen Clothing Company produce each day so as to marimize its daily profit? What is the company's maximum daily profit as a result? Hint: There are actually multiple optimal solutions. List and explain two of them. Would there be a reason for the Shirts Happen Clothing Company to favor one optimal solution over another? Clearly explain your reasoning along with any additional assumptions that need to be made in determining an optimal solution to this problem.wiii-hints Happen": An application in resource allocation and linear programming The Shirts Happen Clothing Company manufactures men's shirts and women's blouses The production process includes cutting. sewing1 and packaging. The table lJEl-W grim the labor re- quirements [in minutes} for each type of garment: Minutes IEI' ;: II 111'. a... semi mass an an 12 an an . The maximum number of labor hours available per day at Shirts Happen are 25 hours for cutting1 35 hours for sewing1 and 5 hours for paelmging. 1. Let 1: represent the number of men's shirts produoed per day by the Shirts Happen Clothing lCompany1 and let 1; represent the number of 1women's blouses produced per day. 'Write a system of Eye linear inequalities involving r and y that, when solved. give the set of all ordered pairs [:r:1 y} of number of men's shirts and number of women's blouses that Ehirts Happen can produce per clay+ given the oonstraints above. Next to each ins-quality1 1write a brief description or interpretation of its meaning. 2. Carefully graph the system of linear inequalitim on. a separate sheet of graph paper and then shade the feasible region. Is it hounded or unbounded? Identify and label on your graph the coordinates of each of the corner points. Attach the graph along with 3mm work. 3. For each of the following solutionsI determine whether or not it is a feasible solution to the Shirts Happen Clothing Company problem. Show your work and explain your reasoning. [a]. 5 men's shirts per day. it} women's blouses per day {h}. 25 men's shirts per day. 25 1women's blouses per day [1:]. I] men's shirts per day. 1] women's blous- per day {d}. i men's shirts per Inlay1 2 women's blouses per day {e}. 10 men's shirts per day. l women's blous- per day {l}. 13 menl's shirts per day, 14 women's blouses per day