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The problem you're working with is related to cost, revenue, and profit for a company producing t-shirts.

### Key information provided:
- **Fixed costs:** 70,000 soles
- **Variable cost per t-shirt:** 21 soles
- **Selling price per t-shirt:** 35 soles

We will need to solve for two things:
1. **Break-even point** (when profit starts, i.e., utility \( U > 0 \)).
2. **Number of units to sell for a profit of at least 2,800 soles**.

### Step-by-step solution:

#### Part (a): Determine the minimum number of units that need to be sold to generate profit.
The company generates profit when the revenue \( R \) is greater than the total costs \( C \).

- **Revenue:** \( R(x) = 35x \), where \( x \) is the number of t-shirts sold.
- **Total cost:** \( C(x) = 70,000 + 21x \) (fixed costs + variable costs).

Profit \( U(x) \) is given by:
\[
U(x) = R(x) - C(x) = 35x - (70,000 + 21x)
\]
Simplifying:
\[
U(x) = 35x - 70,000 - 21x = 14x - 70,000
\]
To find the break-even point (when \( U(x) = 0 \)):
\[
14x - 70,000 = 0
\]
\[
14x = 70,000
\]
\[
x = \frac{70,000}{14} = 5,000
\]

So, the company needs to sell **5,000 units** to start generating profit.

#### Part (b): Determine the number of units to sell to achieve a profit of at least 2,800 soles.
We use the same profit formula:
\[
U(x) = 14x - 70,000
\]
We set \( U(x) \geq 2,800 \):
\[
14x - 70,000 \geq 2,800
\]
\[
14x \geq 2,800 + 70,000
\]
\[
14x \geq 72,800
\]
\[
x \geq \frac{72,800}{14} = 5,200
\]

Thus, the company needs to sell **at least 5,200 units** to have a profit of 2,800 soles or more.

### Final answers:
- (a) The minimum number of units to sell to generate profit is **5,000 units**.
- (b) The minimum number of units to sell to achieve a profit of 2,800 soles or more is **5,200 units**.

Would you like further details or clarifications on any part of the process? Here are some related questions to expand on:

1. How is the break-even point influenced by an increase in fixed costs?
2. How would the profit formula change if the selling price increases?
3. What would happen if the variable cost per unit increased?
4. Can you express the relationship between units sold and profit as a graph?
5. What happens to the required number of units to achieve profit if both fixed and variable costs increase?

**Tip:** When solving profit-related problems, clearly define revenue, fixed costs, and variable costs to set up the equation efficiently.

En una compañía que fabrica camisetas deportivas, el costo por mano de obra y materiales es de S/. 21 por cada camiseta producida mientras sus costos fijos ascienden a S/. 70,000. Si el precio de cada camiseta es de S/. 35, determine la cantidad mínima de unidades que debe vender para que la compañía genere utilidades y la cantidad de unidades para obtener una utilidad de al menos 2,800 soles.

Calculate the minimum number of t-shirts a company must sell to generate profit, given fixed costs of S/. 70,000 and variable costs of S/. 21 per t-shirt. The selling price is S/. 35. Also, find the number of units needed for a profit of 2,800 soles.

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