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Let's work through the problem step by step.

You are selling bottles of water with a cost of $2.09 per bottle and a fixed weekly fee of $250 for the booth. You sell each bottle for $3.39.

### a. Write a function, $ C(n) $, to represent your total costs for the week if you sell $ n $ bottles of water.

The total cost is the sum of the fixed weekly booth fee and the cost per bottle times the number of bottles sold:

\[
C(n) = 250 + 2.09n
\]

Where:
- $ 250 $ is the fixed cost for the booth.
- $ 2.09n $ is the variable cost for $ n $ bottles sold at $2.09 each.

### b. Write a function, $ R(n) $, to represent the revenue from the sale of $ n $ bottles of water during the week.

Revenue is calculated by multiplying the price per bottle by the number of bottles sold:

\[
R(n) = 3.39n
\]

Where:
- $ 3.39n $ is the revenue from selling $ n $ bottles at $3.39 each.

### c. Write a function, $ P(n) $, that represents the profits for selling $ n $ bottles of water in a given week.

Profit is revenue minus cost:

\[
P(n) = R(n) - C(n)
\]
Substituting the functions from parts (a) and (b):
\[
P(n) = 3.39n - (250 + 2.09n)
\]
Simplifying:
\[
P(n) = 3.39n - 250 - 2.09n
\]
\[
P(n) = (3.39 - 2.09)n - 250
\]
\[
P(n) = 1.30n - 250
\]

### d. How many items must you sell to break even?

Break-even occurs when profit $ P(n) = 0 $. So, we set the profit function equal to zero and solve for $ n $:

\[
0 = 1.30n - 250
\]
\[
1.30n = 250
\]
\[
n = \frac{250}{1.30} = 192.31
\]

Since you can't sell a fraction of a bottle, you'll need to sell **193 bottles** to break even.

---

Would you like any further clarification on these steps?

Here are five related questions to explore:
1. What happens to the break-even point if the price per bottle increases?
2. How does reducing the booth fee affect the number of bottles needed to break even?
3. How would the total cost function change if you get a discount for buying in bulk?
4. What would the profit function look like if the cost per bottle fluctuates weekly?
5. How would a sales tax on each bottle affect your profit function?

**Tip**: When determining profit, always compare the rate of revenue increase (price per bottle) with the rate of cost increase to understand how each sale impacts your net profit.

You decide to begin selling bottles of water at the local art festival. Your cost for each bottle of water is $2.09 plus you have to pay a fixed weekly fee of $250 for the booth. Your plan is to sell each bottle of water for $3.39. Write functions for total costs, revenue, and profit, and calculate the break-even point.

Learn how to calculate total costs, revenue, and profit from selling bottles of water. This detailed solution includes break-even analysis for selling $3.39 water bottles with a $250 booth fee and $2.09 cost per bottle. Ideal for high school algebra students.

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