A lunch stand makes $.75 profit on each chef's salad and $1.20 profit on each Caesar salad. On a typical weekday, it sells between 40 and 60 chef's salads and between 35 and 50 Caesar salads. The total number has never exceeded 100 salads. How many of each should be prepared in order to maximize profit?
To maximize profit, we need to find the number of chef's salads and Caesar salads that should be prepared. Let's assume x represents the number of chef's salads and y represents the number of Caesar salads.
Given:
Profit on each chef's salad = $0.75
Profit on each Caesar salad = $1.20
Based on the given information, the number of chef's salads (x) should be between 40 and 60, and the number of Caesar salads (y) should be between 35 and 50. Also, the total number of salads (x + y) should not exceed 100.
Since we want to maximize profit, we need to set up an objective function. Let's say P represents the profit:
P = 0.75x + 1.20y
To find the maximum profit, we need to find the maximum value of P within the given constraints. These constraints are:
40 ≤ x ≤ 60
35 ≤ y ≤ 50
x + y ≤ 100
To solve this problem, we can use a technique called linear programming. We will use the graphical method to find the feasible region and then determine the corner points of the feasible region to evaluate the objective function and find the maximum profit.
First, let's plot the inequalities on a graph to visualize the feasible region.
On the x-axis, plot the range 40 to 60, and on the y-axis, plot the range 35 to 50. Draw lines for the inequalities x = 40, x = 60, y = 35, y = 50, and x + y = 100.
Next, we find the intersection points of these lines to find the feasible region. These points are:
(40, 35), (40, 50), (60, 35), and (50, 50).
Now, substitute each of these corner points into the objective function P = 0.75x + 1.20y to calculate the profit for each corner point:
Corner point (40, 35):
P1 = 0.75(40) + 1.20(35)
Corner point (40, 50):
P2 = 0.75(40) + 1.20(50)
Corner point (60, 35):
P3 = 0.75(60) + 1.20(35)
Corner point (50, 50):
P4 = 0.75(50) + 1.20(50)
Compare these values to find the maximum profit and the corresponding values of x and y.
The combination of chef's salads and Caesar salads that will maximize profit is the one that corresponds to the corner point with the highest profit value.