https://file.aiquickdraw.com/mathbot/file/1728580234727r8sb92yk.jpg

Let's break down the problem and find the solution.

We are given:
- The motivational speaker sells regular and VIP tickets.
- They must sell a **minimum** of:
  - 60 VIP tickets.
  - 120 regular tickets.
- The speaker makes a **profit** of:
  - $225 per regular ticket.
  - $210 per VIP ticket.
- The **total capacity** of the venue is **300 people**.

We want to find out how many of each ticket should be sold to maximize profits.

### Step 1: Define variables
Let:
- $ x $ = number of regular tickets sold.
- $ y $ = number of VIP tickets sold.

### Step 2: Set up the constraints
From the problem:
1. $ x + y = 300 $ (since the total number of tickets is limited to 300).
2. $ x \geq 120 $ (minimum regular tickets sold).
3. $ y \geq 60 $ (minimum VIP tickets sold).

### Step 3: Set up the objective function
The **profit function** to maximize is:
\[
P(x, y) = 225x + 210y
\]
where $ P $ is the total profit.

### Step 4: Solve the system
To maximize the profit while meeting the constraints, let's consider a few combinations:

1. **If the minimum number of each ticket is sold:**
   - $ x = 120 $ and $ y = 60 $.
   - Total tickets = $ 120 + 60 = 180 $, leaving room for $ 300 - 180 = 120 $ more people.

2. **Maximizing regular tickets:**
   If we sell as many regular tickets as possible while meeting the constraints:
   - $ x = 240 $ (since $ x \geq 120 $ and total tickets is 300).
   - $ y = 60 $.
   - Profit = $ 225(240) + 210(60) = 54,000 + 12,600 = 66,600 $.

3. **Maximizing VIP tickets:**
   If we sell as many VIP tickets as possible:
   - $ x = 120 $ and $ y = 180 $.
   - Profit = $ 225(120) + 210(180) = 27,000 + 37,800 = 64,800 $.

### Step 5: Conclusion
The maximum profit occurs when $ 240 $ regular tickets and $ 60 $ VIP tickets are sold, yielding a profit of **$66,600**.

Would you like any more details or have any questions about this solution?

Here are some related questions:
1. What would happen to profits if we increased the venue capacity?
2. How would the solution change if profits per ticket were different?
3. What is the significance of the minimum ticket requirements in the profit calculation?
4. Could we have more than one solution that yields maximum profit?
5. How would the scenario change if the speaker gave discounts on bulk ticket purchases?

**Tip:** Always list the constraints clearly when working on optimization problems; they play a crucial role in shaping your solution.

A motivational speaker sells regular tickets and VIP tickets to their show. In order for the event to be profitable, it must sell a minimum of 60 VIP tickets and a minimum of 120 regular tickets. The speaker makes a profit of $225 for each regular ticket and a profit of $210 for each VIP ticket. The venue has room for 300 people. How many of each ticket should be sold to maximize profits?

Learn how to maximize the profit of a motivational speaker selling regular and VIP tickets, subject to minimum ticket requirements and a venue capacity of 300 people. Solve using linear programming and optimization techniques.

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