The Audubon society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students 56¢ per serving of pasta. The only expenses the society will incur are the cost of the pasta, estimated at 19¢ per serving, and the $370 cost of renting the facility for the evening.
(a)
Write down the associated cost function C(x) in dollars.
C(x) =

Write down the revenue function R(x) in dollars.
R(x) =

Write down the profit function P(x) in dollars.
P(x) =

(b)
How many servings of pasta must the Audubon society sell to break even?
servings
(c)
What profit (or loss

Let's break down the problem step by step.

### Part (a):

#### **Cost Function, $ C(x) $**:
The cost function represents the total expenses incurred by the Audubon Society, which consists of two components:
1. The **fixed cost** of renting the facility for $370.
2. The **variable cost** of making pasta, estimated at 19¢ per serving, or $0.19 per serving.

So, the total cost function is the sum of these two components:
\[
C(x) = 0.19x + 370
\]
where $ x $ is the number of servings of pasta.

#### **Revenue Function, $ R(x) $**:
The revenue function represents the income generated by selling servings of pasta. Since they are charging 56¢ per serving (which is $0.56), the revenue function is:
\[
R(x) = 0.56x
\]
where $ x $ is the number of servings sold.

#### **Profit Function, $ P(x) $**:
The profit is the difference between the revenue and the cost:
\[
P(x) = R(x) - C(x)
\]
Substituting the expressions for $ R(x) $ and $ C(x) $:
\[
P(x) = 0.56x - (0.19x + 370)
\]
Simplifying this:
\[
P(x) = 0.56x - 0.19x - 370
\]
\[
P(x) = 0.37x - 370
\]

### Part (b): Break-even Point
To break even, the profit must be zero, i.e., $ P(x) = 0 $. This gives us:
\[
0.37x - 370 = 0
\]
Solve for $ x $:
\[
0.37x = 370
\]
\[
x = \frac{370}{0.37} = 1000
\]
So, the Audubon Society must sell **1,000 servings** of pasta to break even.

### Part (c): Profit (or loss)
You can now use the profit function to calculate the profit (or loss) based on how many servings of pasta they sell. For example, if they sell more than 1,000 servings, they will make a profit. If they sell fewer than 1,000 servings, they will incur a loss.

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Would you like to explore how much profit (or loss) they'd make for a specific number of servings?

### Five related questions:
1. How much profit would the Audubon Society make if they sold 1,500 servings?
2. What would be the total revenue if 800 servings were sold?
3. What is the cost per serving that would lead to a break-even point of 900 servings?
4. How would the profit function change if the facility rental fee increased to $500?
5. What happens to the break-even point if the price per serving increases to 65¢?

### Tip:
When calculating break-even points, always remember that fixed costs must be covered by the profit margin generated from each unit sold.

The Audubon society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students 56¢ per serving of pasta. The only expenses the society will incur are the cost of the pasta, estimated at 19¢ per serving, and the $370 cost of renting the facility for the evening.
(a)
Write down the associated cost function C(x) in dollars.
C(x) =

Write down the revenue function R(x) in dollars.
R(x) =

Write down the profit function P(x) in dollars.
P(x) =

(b)
How many servings of pasta must the Audubon society sell to break even?
servings
(c)
What profit (or loss) will be incurred for different numbers of servings sold?

Explore how to calculate cost, revenue, and profit functions for a fundraising event at Enormous State University. Learn how to break even by selling pasta at 56¢ per serving, with key functions such as C(x) = 0.19x + 370, R(x) = 0.56x, and P(x) = 0.37x - 370. Suitable for high school students.

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