The Audubon society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students 90¢ per serving of pasta. The only expenses the society will incur are the cost of the pasta, estimated at 18¢ per serving, and the $360 cost of renting the facility for the evening.
(a)
Write down the associated cost function C(x) in dollars.
C(x) = $$  
Correct: Your answer is correct.
 Amazing work!
Write down the revenue function R(x) in dollars.
R(x) = $$  
Correct: Your answer is correct.
 Amazing work.
Write down the profit function P(x) in dollars.
P(x) = $$  
Correct: Your answer is correct.
 Perfect!
(b)
How many servings of pasta must the Audubon society sell to break even?

 servings
(c)
What profit (or loss) results from the sale of 1,500 servings of pasta?

Let's solve the problem step by step:

### Part (a)

#### 1. **Cost Function $ C(x) $**:
The total cost consists of:
- A fixed cost of \$360 for renting the facility.
- A variable cost of 18¢ (or \$0.18) per serving of pasta.

So, the total cost function $ C(x) $ (in dollars) for $ x $ servings of pasta is:
\[
C(x) = 360 + 0.18x
\]

#### 2. **Revenue Function $ R(x) $**:
The society charges 90¢ (or \$0.90) per serving of pasta. The revenue function is:
\[
R(x) = 0.90x
\]

#### 3. **Profit Function $ P(x) $**:
Profit is calculated as the revenue minus the cost. So the profit function is:
\[
P(x) = R(x) - C(x)
\]
Substitute the expressions for $ R(x) $ and $ C(x) $:
\[
P(x) = 0.90x - (360 + 0.18x)
\]
Simplifying the profit function:
\[
P(x) = 0.90x - 360 - 0.18x
\]
\[
P(x) = 0.72x - 360
\]

### Part (b)

To **break even**, the profit must be zero, i.e., $ P(x) = 0 $.

So, we set the profit function equal to zero:
\[
0.72x - 360 = 0
\]
Solve for $ x $:
\[
0.72x = 360
\]
\[
x = \frac{360}{0.72}
\]
\[
x = 500
\]

The society must sell **500 servings** of pasta to break even.

### Part (c)

To find the **profit or loss** from selling 1,500 servings of pasta, we substitute $ x = 1500 $ into the profit function $ P(x) $:
\[
P(1500) = 0.72(1500) - 360
\]
\[
P(1500) = 1080 - 360
\]
\[
P(1500) = 720
\]

The profit from selling 1,500 servings is **\$720**.

---

### Summary:
- Cost function: $ C(x) = 360 + 0.18x $
- Revenue function: $ R(x) = 0.90x $
- Profit function: $ P(x) = 0.72x - 360 $
- Break-even point: **500 servings**
- Profit for 1,500 servings: **\$720**

Would you like more details or have any questions?

### 5 Related Questions:
1. How would the break-even point change if the cost of the pasta increases?
2. What happens to the profit if the price per serving is reduced to 80¢?
3. If the fixed cost was \$500 instead of \$360, how many servings would be needed to break even?
4. How would a 10% increase in student participation affect the profit?
5. What is the loss incurred if only 400 servings are sold?

### Tip:
In profit analysis, understanding the break-even point is crucial—it helps you determine the minimum sales needed to avoid losses.

The Audubon society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students 90¢ per serving of pasta. The only expenses the society will incur are the cost of the pasta, estimated at 18¢ per serving, and the $360 cost of renting the facility for the evening. (a) Write down the associated cost function C(x) in dollars. (b) How many servings of pasta must the Audubon society sell to break even? (c) What profit (or loss) results from the sale of 1,500 servings of pasta?

Learn how to calculate the cost, revenue, and profit functions for the Audubon Society's fundraising event at Enormous State University. This step-by-step solution includes the break-even point and profit analysis for 1,500 servings of pasta. Suitable for Grades 9-12.

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