Have you ever baked cookies from scratch? They require many ingredients in different amounts. Other baked items, such as banana bread, use similar ingredients to make a very different product. Consider these two ingredients lists: Banana Nut Bread Sugar Cookies 2 cups sugar 1 cup butter 4 eggs 3 cups mashed, ripe bananas 1 cup buttermilk 2 teaspoon vanilla 5 cups flour 2 teaspoon baking soda 2 teaspoon salt 2 cup chopped pecans Makes 2 loaves of banana nut bread

1 cups sugar 1 cup butter 1 egg 1 teaspoon vanilla 2 cups flour 1 teaspoon baking soda teaspoon baking powder

Makes 4 dozen sugar cookies

Julia has started a small bakery that will make only these two recipes. She orders 2 pounds of flour. A pound of all-purpose flour contains approximately 18 cups of flour.

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a. Julia is deciding how many batches of sugar cookies and how many loaves of banana bread to make. Construct a system of linear inequalities that shows how many batches of sugar cookies (x) and loaves of banana bread (y) she can make with the flour she has but also reflects the fact that she can't produce negative goods.

tby entering the appropriate corhe area of the graph that includes the inequality. What does each of these features of the graph represent? 1. y-intercept 2. x-intercept 3. shaded area 4. unshaded area 5. points on the line

Task 2: Determining Optimal Profit with Flour Constraints When Julia sells baked goods, she can make a profit of $1.45 per dozen cookies and $4 for a loaf of banana bread. With this information, you can use the graph you constructed of the linear inequality for flour to determine the most profitable use of flour. By examining the vertices of the triangle formed, you can search for the optimal profit for Julia's bakery. a. First find the coordinates for the three corners of the triangle formed by your inequality graph. What does each represent?

b. The profit was given per dozen cookies and per loaf of bread. Convert these profit rates into profit per batch for each recipe.

c. For each point you found in part a, calculate the profit made for those combination of recipes. Note that partial recipes of banana bread cannot be made. Round down the number of batches made when doing your calculations. Type your response here:

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d. If all points in the shaded area are possible combinations of bread and cookies to bake, why do you think the optimal solutions are found only on the edges of the area? Type your response here:

Task 3: Determining Optimal Profit with Multiple Constraints a. Now Julia wishes to take into account more than just flour for her bakery. She has purchased 8 pounds of sugar and 20 sticks of butter. There are about two cups per pound of sugar, and every stick of butter contains about cup of butter. Use the recipes for the two baked goods to construct two inequalities to model the amount of butter and sugar used in these recipes.

b. Open the spreadsheet you've been working with and select the Flour, Sugar, and Butter tab. Enter the equations you've developed into the spreadsheet. For each equation, adjust one of the transparent triangles sitting next to the graph so it fits within underneath one of the lines. The portion of the graph that is darkest should be the area of the graph where all the inequalities overlap. What is the significance of this area? What does each corner of this irregular polygon represent? Copy and paste the graph into this document.

c. Optimal profit can be found by identifying the coordinates of all the corners of the solution region. Name all five points that make up the corners of the solution region on your graph. Type your response here:

d. Assuming that cookies make $1.45 per dozen and bread makes $4 per loaf in profit, calculate the profit earned for each of the potential optimum points on the area. How many batches of bread and cookies should Julia bake?

e. One of the corners of the solution region will always be the answer to the most profitable solution in these types of problems, even though all points within the shaded region satisfy the constraints of the problem. In terms of what is changing at these corners, what do they represent? Why do you think the optimal answer appears at the corners of the answer region instead of at other places on its border?