A local bagel shop produces two products: bagels (B)and croissants (C).Each bagel requires 6 ounces of flour,1 gram of yeast,and 2 tablespoons of sugar.A croissant requires 3 ounces of flour,1 gram of yeast,and 4 tablespoons of sugar.The company has 6,600 ounces of flour,1,400 grams of yeast,and 4,800 tablespoons of sugar available for today's production run.Bagel profits are 20 cents each,and croissant profits are 30 cents each.Using the graphical method,what are optimal profits for today's production run?
A)$580
B)$340
C)$220
D)$380
E)$420

Question

Quizplus · Accepted Answer

Let x be the number of bagels produced and y be the number of croissants produced. The objective function is to maximize profit:
P = 0.20x + 0.30y
The constraints are:
6x + 3y ≤ 6,600 (flour constraint)
1x + 1y ≤ 1,400 (yeast constraint)
2x + 4y ≤ 4,800 (sugar constraint)
x ≥ 0 and y ≥ 0 (non-negativity constraint)
We can graph the feasible region by plotting the boundary lines of each constraint and shading in the region that satisfies all constraints. The feasible region is a polygon with vertices at (0, 0), (600, 0), (400, 800), (933.33, 600), and (1,400). We can evaluate the profit function at each vertex to find the optimal profit:
P(0, 0) = $0
P(600, 0) = $120
P(400, 800) = $380
P(933.33, 600) = $407.33
P(0, 1,400) = $420
Therefore, the optimal profit is $407.33 with 933.33 bagels and 600 croissants produced. However, since we cannot produce a fractional amount of bagels, the actual number produced should be either 933 or 934 bagels with the remaining amount produced as croissants. This would give a profit of $380.

A Local Bagel Shop Produces Two Products: Bagels (B)and Croissants