Question 1184772: A baker produces whole wheat bread and cheese bread to sell at the farmer's market. The whole wheat bread is denser and requires more baking time, where as the cheese bread requires more labor. The oven space restricts the baker from baking more than 120 loaves. Furthermore time for baking is no more than 165 hours and the amount of available labor is at most 55 hours.
Type of bread: Time to Bake: Labor hours: Profit per loaf:
Whole wheat 1.5 hr 1/3 hr 1.20$
Cheese 1 hr 1/2 hr 1.00$
A) How many loaves of each type of bread should be made to maximize the profit? What is that profit?
B)How does the production schedule change if the profit for cheese bread increase to $1.50.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of loaves of whole wheat bread.
y = number of loaves of cheese bread.
objective function:
maximize profit of 2.1 * x + y
constraints:
x,y >= 0
x + y <= 120
1.5x + y <= 165
1/3 * x + 1/2 * y <= 55
with the desmos.com calculator:
graph the opposite of the inequalities.
the region of feasibility is the area of the graph not shaded.
evaluate each corner point of the feasible region with the objective function.
the solution will be at one of the corner points.
the graph looks like this:
the profit at:
(0,110) = 0 * 1.2 + 110 * 1 = 110
(30,90) = 30 * 1.2 + 90 * 1 = 126
(90,30) = 90 * 1.2 + 30 * 1 = 138 *****
(110,0) = 110 * 1.2 + 0 * 1 = 132
the maximum profit is at (90,30)
all the constraints are met.
x,y are both >= 0
x + y = 90 + 30 = 120 <= 120
1.5x + y = 165 <= 165
1/3x + 1/2y = 30 + 15 = 45 <= 55
solution is (x,y) = (90,30) = 138
that means 90 loaves of whole wheat bread and 30 loaves of cheese bread for maximum profit.
when the profit of cheese bread goes from 1 to 1.5, you get:
(0,110) = 0 * 1.2 + 110 * 1.5 = 165
(30,90) = 30 * 1.2 + 90 * 1.5 = 171 *****
(90,30) = 90 * 1.2 + 30 * 1.5 = 153
(110,0) = 110 * 1.2 + 0 * 1.5 = 132
the maximum profit is now at (x,y) = (30,90) = 171.
that means 30 loaves of whole wheat bread and 90 loaves of cheese bread for maximum profit.
the graph is the same because none of the constraints has changed.
the only change is the evaluation of the objective function at each of the corner points.
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