The owner of Crackers, Inc. produces both Deluxe (D) and Classic (C) crackers. She only has 4,800 ounces of sugar, 9,600 ounces of flour, and 2,000 ounces of salt for her next production run. A box of Deluxe crackers requires 2 ounces of sugar, 6 ounces of flour, and 1 ounce of salt to produce. A box of Classic crackers requires 3 ounces of sugar, 8 ounces of flour, and 2 ounces of salt to produce. Profits are 40 cents for a box of Deluxe crackers and 50 cents for a box of Classic crackers. Cracker's, Inc. would like to maximize profits.
What is the daily profit when producing the optimal amounts?
A) $800
B) $500
C) $640
D) $620
E) $600

Question

Quizplus · Accepted Answer

Let x be the number of boxes of Deluxe crackers and y be the number of boxes of Classic crackers produced per day.
The objective is to maximize profit, which is given by P = 0.40x + 0.50y.
The constraints are as follows:
2x + 3y ≤ 4,800 (sugar constraint)
6x + 8y ≤ 9,600 (flour constraint)
x + 2y ≤ 2,000 (salt constraint)
x, y ≥ 0 (non-negativity constraints)
Rewriting the first constraint as y ≤ (4,800 - 2x) / 3 and graphing it, we get:
[graph goes here]
Similarly, plotting the second and third constraints on the same graph, we get:
[graph goes here]
The feasible region is the shaded triangle bounded by the three constraints and the axes. We need to find the corner points of this triangle and evaluate the objective function at each of them to determine the optimal solution.

Corner point 1: x = 0, y = 0, P = 0
Corner point 2: x = 0, y = 1, P = 0.50
Corner point 3: x = 800, y = 0, P = 320
Corner point 4: x = 1,200, y = 400, P = 620
Corner point 5: x = 600, y = 800/3, P = 500

The maximum profit of $640 is obtained when 1,200 boxes of Deluxe crackers and 400 boxes of Classic crackers are produced per day. Therefore, the answer is (C).

The Owner of Crackers, Inc