A furniture manufacturing company spends ₱50 000 per
month for its operation. They also spend ₱4 000 for each
cabinet they produce. If the company sells a cabinet for
₱8 000 a piece, determine the function that represents how
much profit they earned per cabinet and identify if the function
is rational.

Let's break down the given problem and establish the profit function.

### Key Information:
- **Fixed monthly cost**: ₱50,000 (this is the operation cost, which does not depend on the number of cabinets produced).
- **Cost per cabinet**: ₱4,000.
- **Selling price per cabinet**: ₱8,000.

### Let:
- \( x \) = number of cabinets produced and sold.

### Total Cost:
- The total cost consists of the fixed monthly cost and the variable cost for producing \( x \) cabinets.
  \[
  \text{Total Cost} = 50,000 + 4,000x
  \]

### Total Revenue:
- The total revenue comes from selling \( x \) cabinets at ₱8,000 each.
  \[
  \text{Total Revenue} = 8,000x
  \]

### Profit Function:
Profit is calculated as the difference between total revenue and total cost.
\[
P(x) = \text{Total Revenue} - \text{Total Cost}
\]
Substituting the expressions for total revenue and total cost:
\[
P(x) = 8,000x - (50,000 + 4,000x)
\]
Simplifying:
\[
P(x) = 8,000x - 50,000 - 4,000x
\]
\[
P(x) = 4,000x - 50,000
\]

Thus, the **profit function** is:
\[
P(x) = 4,000x - 50,000
\]

### Is the function rational?

A **rational function** is one that can be expressed as the ratio of two polynomials. In this case, the profit function \( P(x) = 4,000x - 50,000 \) is a linear function (a polynomial of degree 1). It is not in the form of a ratio (quotient) of two polynomials, so **this is not a rational function**. It is a **linear function**.

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Would you like further details or have any questions?

### Related questions:
1. How would the profit change if the cost per cabinet increased?
2. What is the breakeven point for the company?
3. How does the profit function behave if the fixed cost increases?
4. Can this be represented graphically, and what would the slope indicate?
5. How much should they sell the cabinet for to double their profits?

**Tip**: To increase profitability, reducing variable costs or increasing the selling price can have a significant impact.

A furniture manufacturing company spends ₱50 000 per month for its operation. They also spend ₱4 000 for each cabinet they produce. If the company sells a cabinet for ₱8 000 a piece, determine the function that represents how much profit they earned per cabinet and identify if the function is rational.

Learn how to calculate the profit function for a furniture manufacturing company. This problem involves linear functions and profit calculations, using key concepts such as fixed costs, variable costs, and revenue. Suitable for students in Grades 9-10.

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