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**Question 7 (5 pts)**

The monthly cost, in hundreds of dollars, to produce \(x\) Weekly Advanced Fun Packs (WA) is given by the cost function \( C(x) = 3x + 1000 \), and the price-demand function, in hundreds of dollars per Fun Pack, is given by \( p(x) = 140 - 2x \), where \( x \leq 70 \).

1. **Find the Profit function.**
2. **Find the minimum number of guitar packs required to break even.**
3. **Find the marginal profit function and use it to determine the maximum profit and when it occurs. Summarize your results in a sentence.**

For full credit, you must submit your work with your PDF submission in Exam 3.

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I can help you with solving each part of the question. Let's start with part 1:

### 1. Find the Profit function:

Profit \( P(x) \) is defined as:

\[
P(x) = R(x) - C(x)
\]

Where \( R(x) \) is the revenue function and \( C(x) \) is the cost function.

- The **revenue function** is given by \( R(x) = p(x) \cdot x = (140 - 2x) \cdot x = 140x - 2x^2 \).
- The **cost function** is given as \( C(x) = 3x + 1000 \).

Now, the profit function is:

\[
P(x) = (140x - 2x^2) - (3x + 1000)
\]
\[
P(x) = 140x - 2x^2 - 3x - 1000
\]
\[
P(x) = 137x - 2x^2 - 1000
\]

So, the **Profit function** is:

\[
P(x) = -2x^2 + 137x - 1000
\]

Would you like to proceed to part 2 or need further clarification on part 1?

The monthly cost, in hundreds of dollars, to produce x Weekly Advanced Fun Packs (WA) is given by the cost function C(x) = 3x + 1000, and the price-demand function, in hundreds of dollars per Fun Pack, is given by p(x) = 140 - 2x, where x <= 70. Find the Profit function, the minimum number of guitar packs required to break even, and the marginal profit function to determine the maximum profit and when it occurs.

Explore the derivation of the profit function, how to calculate the break-even point, and the analysis of marginal profit using algebra and basic economic concepts. This example uses linear cost and quadratic revenue functions to model real-world business scenarios in a college-level economics or business course.

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