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Let's solve the three parts based on the information provided in the image:

### Given Data:
- Fixed costs: $1300 per month
- Variable cost per T-shirt: $6
- Selling price per T-shirt: $10

#### (a) Cost Function $ C(t) $:
The total cost consists of fixed costs and variable costs. The fixed cost is $1300 per month, and the variable cost per T-shirt is $6. The cost function can be written as:

\[
C(t) = 1300 + 6t
\]

where $ t $ is the number of T-shirts produced.

#### (b) Revenue Function $ R(t) $:
The revenue is obtained by selling $ t $ T-shirts at $10 each. Therefore, the revenue function is:

\[
R(t) = 10t
\]

#### (c) Break-even Point:
The break-even point occurs when the total cost equals the total revenue, i.e., when $ C(t) = R(t) $. Setting the two functions equal to each other:

\[
1300 + 6t = 10t
\]

Now, solve for $ t $:

\[
1300 = 10t - 6t
\]

\[
1300 = 4t
\]

\[
t = \frac{1300}{4} = 325
\]

Thus, the break-even point is $ t = 325 $. This means the manufacturer needs to produce and sell 325 T-shirts to break even.

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Here are 5 related questions:
1. What happens to the profit if more than 325 T-shirts are sold?
2. How would the cost function change if the fixed costs increase?
3. What is the profit function based on the cost and revenue functions?
4. How many T-shirts need to be sold to make a $500 profit?
5. How does the variable cost per T-shirt affect the break-even point?

**Tip:** Break-even analysis is important for determining how many units you need to sell to cover your costs.

A manufacturer of T-shirts has fixed costs of $1300 per month and variable costs of $6 per T-shirt. T-shirts sell for $10 each. Find the cost function, revenue function, and break-even point.

Learn how to calculate the cost function, revenue function, and break-even point for a T-shirt manufacturer with fixed costs of $1300 per month and variable costs of $6 per T-shirt. Explore linear equations and break-even analysis in this real-world application suitable for high school students.

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