Your school is organizing a sports festival, and you need to raise P15,000 to support the event. You will decide to sell one type of customized school merchandise (e.g., T-shirts, caps, or water bottles) to students and staff. Your task is to create a mathematical model to predict how much money can be raised based on the number of items sold, the cost of production, and the selling price. You need to determine the number of items to be sold to reach the fundraising goal. and the number of item sold is 200, cost of production is 250 and the selling price is 325. Solve it using function

To solve the problem using a mathematical model, we'll define the following:

- Let \( n \) be the number of items sold.
- Let \( c \) be the cost of production per item.
- Let \( p \) be the selling price per item.
- Let \( R(n) \) be the total revenue generated from selling \( n \) items.
- Let \( C(n) \) be the total cost of production for \( n \) items.
- Let \( P(n) \) be the profit made from selling \( n \) items.

Given:
- Fundraising goal: P15,000
- \( n = 200 \)
- \( c = 250 \)
- \( p = 325 \)

### Step 1: Define the revenue and cost functions.
1. **Revenue Function \( R(n) \)**: This is the total money received from selling \( n \) items.
   \[
   R(n) = p \times n
   \]

2. **Cost Function \( C(n) \)**: This is the total cost of producing \( n \) items.
   \[
   C(n) = c \times n
   \]

3. **Profit Function \( P(n) \)**: This is the profit earned, which is the revenue minus the cost.
   \[
   P(n) = R(n) - C(n)
   \]

### Step 2: Calculate the specific values for the given number of items sold (\( n = 200 \)).

1. **Revenue Calculation**:
   \[
   R(200) = 325 \times 200 = 65,000 \text{ PHP}
   \]

2. **Cost Calculation**:
   \[
   C(200) = 250 \times 200 = 50,000 \text{ PHP}
   \]

3. **Profit Calculation**:
   \[
   P(200) = R(200) - C(200) = 65,000 - 50,000 = 15,000 \text{ PHP}
   \]

### Step 3: Determine the number of items needed to meet the fundraising goal.

To find the number of items \( n \) needed to reach a profit of P15,000, set \( P(n) = 15,000 \) and solve for \( n \).

1. **Profit Equation**:
   \[
   P(n) = (p \times n) - (c \times n) = 15,000
   \]
   Simplify this equation:
   \[
   (325 - 250) \times n = 15,000
   \]
   \[
   75 \times n = 15,000
   \]
   \[
   n = \frac{15,000}{75} = 200
   \]

### Conclusion:
To raise P15,000, you need to sell 200 items, assuming each item costs 250 PHP to produce and is sold for 325 PHP.

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

Here are five related questions:
1. What would happen if the selling price is increased by 10%?
2. How does the profit change if the production cost decreases by 50 PHP per item?
3. What if only 150 items are sold? How much more do you need to sell to reach the goal?
4. What would be the impact on profit if a discount of 10% is offered on the selling price?
5. How many items must be sold if the cost of production increases by 20%?

**Tip**: When planning a fundraiser, always consider fixed costs (if any) and how changes in variable costs or pricing can impact the overall profit.

Learn how to calculate the fundraising goal by selling customized school merchandise. This problem involves determining the number of items to sell based on production costs, selling price, and profit analysis. Solve for the number of items required to raise P15,000.

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