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The problem you're working on is related to linear applications and cost functions. Let's break down the given information:

**Problem Statement:**

- Joanne sells silk-screened T-shirts at community festivals.
- Her marginal cost (cost to produce one T-shirt) is $4.50.
- Her total cost to produce 40 T-shirts is $230.
- She sells each T-shirt for $8.

We are asked to find:

1. The **linear cost function** for Joanne’s T-shirt production.
2. The number of T-shirts she must produce and sell to **break even**.
3. The number of T-shirts she must sell to make a **profit of $5000**.

---

### Solution Outline:

#### a. **Linear Cost Function**
The linear cost function is generally of the form:

\[
C(x) = mx + b
\]

Where:
- $ m $ is the **marginal cost** per T-shirt ($4.50),
- $ b $ is the **fixed cost** (costs that don’t depend on the number of T-shirts produced).

From the problem, we know:
- For $ x = 40 $ (40 T-shirts), $ C(40) = 230 $.
- $ m = 4.50 $.

We can use this to find the fixed cost $ b $:

\[
230 = 4.50(40) + b
\]

Solve for $ b $.

#### b. **Break-Even Point**
The break-even point occurs when **total revenue equals total cost**. The revenue function is:

\[
R(x) = 8x
\]

Set revenue equal to cost:

\[
8x = 4.50x + b
\]

Solve for $ x $ (the number of T-shirts to break even).

#### c. **Profit of $5000**
To find the number of T-shirts she needs to sell for a $5000 profit, use the profit formula:

\[
\text{Profit} = R(x) - C(x)
\]

Set profit equal to 5000:

\[
5000 = 8x - (4.50x + b)
\]

Solve for $ x $ (the number of T-shirts).

---

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1. What is the

Joanne sells silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one T-shirt is $4.50. Her total cost to produce 40 T-shirts is $230, and she sells them for $8 each. Find the linear cost function for Joanne's T-shirt production. How many T-shirts must she produce and sell in order to break even? How many T-shirts must she produce and sell to make a profit of $5000?

Learn how to calculate the cost function for T-shirt production, determine the break-even point, and solve for the number of T-shirts needed to achieve a $5000 profit. This problem covers key concepts in linear functions, revenue, and profit calculation. Suitable for high school students.

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