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### Problem Overview:
Emmanuel sells dairy chocolate at **₱135 per pack**, with a production cost of **₱90 per pack** and operating expenses of **₱4,500**.

### 1. **Which of the following pertains to the profit function?**
The profit function is calculated as:

\[
\text{Profit} = (\text{Selling Price} - \text{Cost Price}) \times x - \text{Operating Expenses}
\]

Where \(x\) is the number of chocolate packs sold.

\[
P(x) = (135 - 90) \times x - 4500 = 45x - 4500
\]

Thus, the correct answer is:

- **a.** \( P(x) = 45x - 4,500 \)

---

### 2. **How many packs of dairy chocolate must be sold to break even?**
Break-even occurs when the profit is zero:

\[
P(x) = 0 \implies 45x - 4500 = 0
\]

Solving for \(x\):

\[
45x = 4500 \implies x = \frac{4500}{45} = 100
\]

The break-even point is **100 packs**. Thus, the correct answer is:

- **b.** 100

---

### 3. **Which value of \(x\) will make Emmanuel's shop suffer a loss?**
Emmanuel suffers a loss when the profit is less than zero:

\[
P(x) < 0 \implies 45x - 4500 < 0
\]

Solving for \(x\):

\[
45x < 4500 \implies x < 100
\]

So, if he sells fewer than 100 packs, he will suffer a loss. Thus, the correct answer is:

- **b.** \( x \leq 100 \)

---

### 4. **How many chocolate bars must be sold for a profit of ₱6,750?**

We set the profit function equal to 6,750:

\[
P(x) = 45x - 4500 = 6750
\]

Solving for \(x\):

\[
45x = 11250 \implies x = \frac{11250}{45} = 250
\]

Thus, Emmanuel must sell **250 packs**. The correct answer is:

- **c.** 250

---

### 5. **How much is the gain if Emmanuel sold 350 packs of chocolates?**

We substitute \(x = 350\) into the profit function:

\[
P(350) = 45(350) - 4500 = 15750 - 4500 = 11,250
\]

Thus, the gain is **₱11,250**. The correct answer is:

- **c.** ₱11,250.00

---

Would you like more detailed explanations for any of these steps? Here are some follow-up questions:

1. What is the definition of a break-even point in business math?
2. How is a profit function derived from a cost function?
3. How can Emmanuel calculate his total revenue if he knows the number of packs sold?
4. What is the formula for calculating loss in a business scenario?
5. How do changes in fixed and variable costs affect the break-even point?

**Tip:** To increase profit, businesses can focus on either increasing the selling price or reducing production and operating costs.

Emmanuel decided to put up a candy shop that sells dairy chocolate that costs ₱135.00 per pack. The cost of making the chocolate is ₱90.00 and the operating expense is ₱4,500.00. Use this information to answer five questions about profit, break-even points, and more.

Solve a profit and break-even analysis problem where Emmanuel sells dairy chocolate at ₱135.00 per pack with a production cost of ₱90.00 and operating expenses of ₱4,500.00. Learn how to calculate profit, determine the break-even point, and analyze loss scenarios using algebraic functions.

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