Bakery has bought 250 pounds of muffin dough. They want to make waffles or muffins in half-dozen packs out of it. Half a dozen of muffins requires 1 lb of dough and a pack of waffles uses 3/4 lb of dough. It take bakers 6 minutes to make a half-dozen of waffles and 3 minutes to make a half-dozen of muffins. Their profit will be $1.50 on each pack of waffles and $2.00 on each pack of muffins. How many of each should they make to maximize profit, if they have just 20 hours to do everything?
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To maximize profit, we need to determine the number of packs of waffles and muffins to make given the constraints.
Let's denote:
x = number of packs of waffles
y = number of packs of muffins
From the problem statement, we can write the following equations:
1. The total weight of dough used should not exceed 250 pounds:
(3/4)x + y <= 250
2. The total time used to make waffles and muffins should not exceed 20 hours:
(6/60)x + (3/60)y <= 20
3. The profit from selling waffles:
Profit from waffles = 1.50x
4. The profit from selling muffins:
Profit from muffins = 2.00y
We want to maximize the total profit (P):
P = Profit from waffles + Profit from muffins
P = 1.50x + 2.00y
To solve this problem, we will use linear programming. We need to find the values of x and y that maximize P while satisfying the constraints.
First, let's convert the time constraint to minutes for consistency. Then we can graph the feasible region on a graph and find the highest point, which represents the maximum profit.
Let's solve the linear programming problem graphically:
1. Plot the feasible region:
Graph the inequality (3/4)x + y <= 250
Graph the inequality (6/60)x + (3/60)y <= 20
These two inequalities will form the boundaries of the feasible region on the graph.
2. Find the vertices of the feasible region:
To find the vertices, solve the system of equations formed by the boundary lines of the feasible region. The vertices are the intersection points of the boundary lines.
3. Substitute the vertices into the profit equation:
Calculate the profit at each vertex by substituting the x and y values into the profit equation P = 1.50x + 2.00y.
4. Determine the maximum profit:
Compare the profits obtained at each vertex and select the highest value as the maximum profit. Note the corresponding values of x and y.
5. Finalize the solution:
Report the values of x and y that maximize profit. These values represent the number of packs of waffles and muffins, respectively.
By following these steps, you can find the solution to maximize profit given the constraints provided.