Apply Polya's problem-solving method to these questions, employing functions for modeling and solving them.

2. A bakery sells cakes for special occasions. The cost to make each cake is 200 pesos, and they are being sold for 500 pesos each. The bakery has already spent 5000 pesos on equipment. How many candles must be made and sold to break even?

To solve this problem using **Polya's problem-solving method**, we will break it down step by step:

### Step 1: Understand the Problem
We need to determine how many cakes the bakery must sell to cover both its variable costs (production cost per cake) and fixed costs (equipment cost) so that it breaks even. The break-even point is when the total revenue equals the total costs.

- **Fixed cost**: The bakery has spent 5000 pesos on equipment.
- **Variable cost**: The cost to make each cake is 200 pesos.
- **Revenue per cake**: The bakery sells each cake for 500 pesos.

### Step 2: Devise a Plan
We can set up a function to model both the costs and the revenue.

1. **Cost Function**:
   The total cost is composed of the fixed cost and the variable cost. If \( x \) represents the number of cakes made and sold, then the total cost \( C(x) \) is:
   \[
   C(x) = 5000 + 200x
   \]
   where:
   - 5000 is the fixed cost,
   - 200x is the variable cost for \( x \) cakes.

2. **Revenue Function**:
   The revenue is the money earned from selling the cakes. If \( x \) is the number of cakes sold, then the revenue \( R(x) \) is:
   \[
   R(x) = 500x
   \]
   where:
   - 500 is the selling price per cake,
   - \( x \) is the number of cakes sold.

3. **Break-even Point**:
   The bakery breaks even when the total cost equals the total revenue. Therefore, we set the cost function equal to the revenue function:
   \[
   C(x) = R(x)
   \]
   Substituting the expressions for \( C(x) \) and \( R(x) \):
   \[
   5000 + 200x = 500x
   \]

### Step 3: Carry Out the Plan
Now, solve the equation for \( x \).

1. Subtract \( 200x \) from both sides:
   \[
   5000 = 500x - 200x
   \]
   \[
   5000 = 300x
   \]

2. Divide both sides by 300 to isolate \( x \):
   \[
   x = \frac{5000}{300}
   \]
   \[
   x \approx 16.67
   \]

Since the bakery cannot sell a fraction of a cake, we round up to the nearest whole number, \( x = 17 \).

### Step 4: Review the Solution
To break even, the bakery must sell **17 cakes**. This is the minimum number of cakes required to cover both the production costs and the equipment investment.

### Conclusion:
The bakery must make and sell 17 cakes to break even.

---

Would you like more details or further clarification on this solution?

Here are 5 questions that expand on the above problem:
1. What if the cost to make each cake increased to 250 pesos? How many cakes would be needed to break even?
2. How would the break-even point change if the bakery sold cakes at a discount of 450 pesos each?
3. If the bakery spent 7000 pesos on equipment instead of 5000, how many cakes must be sold to break even?
4. How much profit does the bakery make after selling 25 cakes?
5. How can we modify the function to include a discount for bulk orders of cakes?

**Tip**: When solving break-even problems, always differentiate between fixed and variable costs to correctly set up the cost function.

Apply Polya's problem-solving method to these questions, employing functions for modeling and solving them.

2. A bakery sells cakes for special occasions. The cost to make each cake is 200 pesos, and they are being sold for 500 pesos each. The bakery has already spent 5000 pesos on equipment. How many cakes must be made and sold to break even?

Learn how to determine the break-even point for a bakery selling cakes, using Polya's problem-solving method. This example covers algebraic concepts, cost and revenue functions, and break-even analysis, suitable for students in Grades 9-12. Find out how many cakes must be sold to cover production and equipment costs.

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