The ticket office makes $5 profit each evening ticket sold and $3 on each matinee ticket sold. The ticket office wants to sell at least 50 evening tickets per week and at least 40 matinee tickets per week. Its goal is to earn at least $250 profit in sales per week. Find all possible combinations of evening tickets and matinee tickets that the ticket office needs to sell to meet their goal.
To find all possible combinations of evening tickets and matinee tickets that the ticket office needs to sell to meet their goal, we can use a systematic approach.
Let's create a table to organize the information:
| Evening Tickets (x) | Matinee Tickets (y) | Evening Profit (5x) | Matinee Profit (3y) | Total Profit (5x + 3y) |
|---------------------|---------------------|---------------------|---------------------|-----------------------|
We know that the ticket office wants to sell at least 50 evening tickets per week and at least 40 matinee tickets per week. Let's start filling in the table with these minimum values.
| Evening Tickets (x) | Matinee Tickets (y) | Evening Profit (5x) | Matinee Profit (3y) | Total Profit (5x + 3y) |
|---------------------|---------------------|---------------------|---------------------|-----------------------|
| 50 | 40 | 250 | 120 | 370 |
Now, we need to find other combinations that result in a total profit of at least $250.
One approach is to increment the number of evening tickets sold while keeping the number of matinee tickets constant at the minimum value of 40, and observe the resulting total profit. We can continue this process until the total profit exceeds or equals $250.
| Evening Tickets (x) | Matinee Tickets (y) | Evening Profit (5x) | Matinee Profit (3y) | Total Profit (5x + 3y) |
|---------------------|---------------------|---------------------|---------------------|-----------------------|
| 50 | 40 | 250 | 120 | 370 |
| 51 | 40 | 255 | 120 | 375 |
| 52 | 40 | 260 | 120 | 380 |
| 53 | 40 | 265 | 120 | 385 |
| ... | ... | ... | ... | ... |
Similarly, we can increment the number of matinee tickets sold while keeping the number of evening tickets constant at the minimum value of 50, and observe the resulting total profit.
| Evening Tickets (x) | Matinee Tickets (y) | Evening Profit (5x) | Matinee Profit (3y) | Total Profit (5x + 3y) |
|---------------------|---------------------|---------------------|---------------------|-----------------------|
| 50 | 40 | 250 | 120 | 370 |
| 50 | 41 | 250 | 123 | 373 |
| 50 | 42 | 250 | 126 | 376 |
| 50 | 43 | 250 | 129 | 379 |
| ... | ... | ... | ... | ... |
Continue this process until you have explored all possible combinations of evening tickets and matinee tickets. Each combination that results in a total profit greater than or equal to $250 will be a valid solution to meet the ticket office's goal.
Please note that as this is a manual iterative process, I have only shown a few steps for illustration. In practice, you would need to perform all the necessary iterations to find all possible combinations.
Let's use a system of inequalities to find all the possible combinations of evening and matinee tickets that meet the ticket office's profit goal.
Let's assume the number of evening tickets sold per week is represented by the variable "E", and the number of matinee tickets sold per week is represented by the variable "M".
Based on the given information, we can form the following inequalities:
1. Profit from evening tickets: 5E (as each evening ticket sold brings in a $5 profit)
2. Profit from matinee tickets: 3M (as each matinee ticket sold brings in a $3 profit)
3. Minimum number of evening tickets: E ≥ 50
4. Minimum number of matinee tickets: M ≥ 40
5. Minimum profit goal: 5E + 3M ≥ 250
Now, let's solve this system of inequalities by graphing:
First, let's consider the inequality E ≥ 50. This represents a horizontal line on the graph, parallel to the x-axis, passing through the point (50, 0).
Next, let's graph the inequality M ≥ 40. This represents a vertical line on the graph, parallel to the y-axis, passing through the point (0, 40).
Finally, let's graph the inequality 5E + 3M ≥ 250. To do this, we can find points that satisfy this inequality, such as (0, 83.33), (50, 0), and (25, 41.67). Drawing a line passing through these points will represent the inequality 5E + 3M ≥ 250.
The shaded region where all three inequalities overlap represents the valid combinations of evening and matinee tickets that meet the ticket office's profit goal.
Please note that the graph is not easily displayed in a text-based format. It's recommended to use graphing software or a graphical calculator to visualize the solution region.