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Problem 1: Manufacturing Costs Function

Statement:
A furniture manufacturing company spends P30,000 per month for its operation. They also spend P3,000 for each cabinet they produce. If the company sells $x$ cabinets for P8,000 each, determine the function that represents how much profit they earned per cabinet and identify if the function is a rational function.

Solution:

1. Fixed Cost: The fixed operational cost is $P30,000$ per month.
2. Variable Cost per Cabinet: The variable cost is $P3,000$ per cabinet.
3. Revenue per Cabinet: They sell each cabinet for $P8,000$.

The total cost $C(x)$ for producing $x$ cabinets is: $C(x) = 30000 + 3000x$

The total revenue $R(x)$ from selling $x$ cabinets is: $R(x) = 8000x$

Profit Function:
Profit $P(x)$ is the difference between revenue and cost: $P(x) = R(x) - C(x) = 8000x - (30000 + 3000x)$

Simplifying this: $P(x) = 8000x - 30000 - 3000x$ $P(x) = 5000x - 30000$

So, the function representing the profit per cabinet is: $P(x) = 5000x - 30000$

Is it a Rational Function?
No, this is not a rational function because a rational function has the form $\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are polynomials. This function is linear.

Problem 2: Average Cost per Dance Lesson

Statement:
To join a dance lesson, you paid a one-time membership fee of P3,000, then P500 for every session. If they give you one free session after every ten sessions and you have attended a number of paid sessions that is a multiple of 10, determine the function that represents the average cost per session and identify if the function is a rational function.

Solution:

1. Fixed Membership Fee: P3,000.
2. Cost per Paid Session: P500.
3. Number of Paid Sessions: Let $n$ be the number of paid sessions.

Total Cost:
Since every 10 sessions give you 1 free session, for $n$ paid sessions, the total number of sessions including free ones is $n + \frac{n}{10}$.

The total cost $T(n)$ is: $T(n) = 3000 + 500n$

Total Number of Sessions:
$S(n) = n + \frac{n}{10} = \frac{11n}{10}$

Average Cost per Session Function:
The average cost per session $A(n)$ is: $A(n) = \frac{T(n)}{S(n)} = \frac{3000 + 500n}{\frac{11n}{10}}$

Simplifying: $A(n) = \frac{10(3000 + 500n)}{11n} = \frac{30000 + 5000n}{11n} = \frac{5000(n + 6)}{11n}$

Is it a Rational Function?
Yes, this is a rational function because it is in the form of $\frac{f(x)}{g(x)}$ where both $f(x)$ and $g(x)$ are polynomials.

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Would you like any more details or have any questions?

Here are 5 related questions to explore further:

1. What is the break-even point for the manufacturing company where profit equals zero?
2. How does increasing the fixed cost affect the profit function in Problem 1?
3. What happens to the average cost per session if the membership fee increases in Problem 2?
4. How would you modify the function in Problem 2 if there were more frequent free sessions?
5. What are the characteristics of a rational function, and how can you identify them?

Tip: When analyzing cost functions, always separate fixed and variable costs to clearly see how each contributes to the overall function.