QUESTION 1: Graphical Method and Sensitivity Analysis (20 points) The graphs of three greater than or equal to constraints are shown below. The arrow for each constraint points to the side of the constraint line that satisfies the constraint. The objective function is 6x+3y. The dotted line is the line 6x+3y=600. For cases (a) and (b) below, describe the outcome of the optimization problem. You do not have to calculate the coordinates of any optimal solutions if you can point to them on the graph OR describe the solutions. Be sure to label your work well. Y 200 100 X 100 200 (a) The objective function is to be maximized. (3 points) (b) The objective function is to be minimized. (3 points) (c) Let Xi = gallons of component used in gasoline j, where i= 1, 2 and j= 1, 2. (Assume that we have two components and two types of gasoline). Write the constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1. (2 points) Mary Custard's is a pie shop that specializes in custard and fruit pies. It makes delicious pies and sells them at reasonable prices so that it can sell all the pies it makes in a day. Every dozen custard pies (12 custard pies) nets Mary Custard's $15 and requires 12 pounds of flour, 50 eggs, 5 pounds of sugar and no fruit mixture. Every dozen of fruit pies (12 fruit pies) nets a $25 profit and uses 10 pounds of flour, 40 eggs, 10 pounds of sugar, and 15 pounds of fruit mixture. On a given day, the bakers at Mary Custard's found that they had 150 pounds of flour, 500 eggs, 90 pounds of sugar and 120 pounds of fruit mixture with which to make pies. The problem when formulated as a linear program and solved is as follows: Max. Z=$15X1 + 25X2 Subject to ci 12X1 + 10X2 = 0 B E F G . | 1 X1 4 2/3 2 Solutions 3 4 Net Profit D X2 6 2/3 Total $25 236 2/3 $15 5 6 12 7 50 C1 C2 C3 C4 10 40 10 122 2/3