The motivational speaker should sell 180 regular tickets and 120 VIP tickets to maximize their profits while fulfilling the minimum ticket requirements and staying within the venue's capacity.
How can you solve how many of each ticket should be sold to maximize profits?
Let:
x = number of regular tickets sold
y = number of VIP tickets sold
2. Formulate the objective function:
We want to maximize the total profit. Therefore, the objective function is:
P(x, y) = 225x + 210y
Minimum VIP tickets: y ≥ 80
Minimum regular tickets: x ≥ 120
Maximum capacity: x + y ≤ 300
This is a linear programming problem. Using the analytical methods to find the feasible region and the point within it that maximizes the objective function.
we convert the constraints into equalities by introducing slack variables.
Formulate the Lagrangian function by adding the objective function and Lagrange multipliers multiplied by the constraints.
Take partial derivatives of the Lagrangian function with respect to x, y, and the Lagrange multipliers, and set them equal to zero.
Solving the system of equations obtained to find the optimal values of x, y, and the Lagrange multipliers.
Solution:
x = 180 regular tickets
y = 120 VIP tickets
Total profit: P(180, 120) = 225 * 180 + 210 * 120 = $63,900
Therefore, the motivational speaker should sell 180 regular tickets and 120 VIP tickets to maximize their profits while fulfilling the minimum ticket requirements and staying within the venue capacity.