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The problem presents a scenario where a company produces and sells 75 items at a price of $50 per item. The cost function for producing these items is given by:

\[
C_t = 30(x) + 1500
\]

Where:
- $x$ is the number of items produced,
- $C_t$ represents the total cost.

We are asked to find the three equations for:
1. Income (Ingreso),
2. Costs (Costos),
3. Profit (Utilidad).

### Step 1: Income Equation (Ingreso)
The income $I(x)$ is the total revenue generated from selling $x$ items at $50 each.

\[
I(x) = 50x
\]

### Step 2: Cost Equation (Costos)
The cost equation is already given as:

\[
C_t(x) = 30x + 1500
\]

This shows that there is a variable cost of $30 per item and a fixed cost of $1500.

### Step 3: Profit Equation (Utilidad)
The profit $U(x)$ is the difference between the income and the cost:

\[
U(x) = I(x) - C_t(x)
\]

Substituting the values of $I(x)$ and $C_t(x)$:

\[
U(x) = 50x - (30x + 1500)
\]

Simplifying:

\[
U(x) = 50x - 30x - 1500
\]

\[
U(x) = 20x - 1500
\]

### Final Equations:
1. Income (Ingreso): $ I(x) = 50x $
2. Cost (Costos): $ C_t(x) = 30x + 1500 $
3. Profit (Utilidad): $ U(x) = 20x - 1500 $

Since the company produces 75 items, let's substitute $ x = 75 $ into these equations to find the specific values for income, cost, and profit.

Let me calculate these values.For the production of 75 items:

1. **Income (Ingreso)**: $ I(75) = 3750 $ dollars.
2. **Cost (Costos)**: $ C_t(75) = 3750 $ dollars.
3. **Profit (Utilidad)**: $ U(75) = 0 $ dollars.

This means that at the production level of 75 items, the company breaks even with no profit or loss.

Would you like further details or clarification on any part of this solution?

Here are 5 related questions to deepen your understanding:
1. How would the profit change if the selling price per item increased?
2. What would happen to the profit if the fixed costs increased?
3. How many items need to be produced for the company to start making a profit?
4. What is the break-even point in this scenario?
5. How would the equations change if the cost per item decreased?

**Tip:** Understanding how to adjust the selling price and costs helps you optimize profit in business scenarios.

Una empresa produce y vende 75 artículos de fantasía, para elaborarlos emplea la siguiente función de costo Ct = 30(x) + 1500, siempre y cuando el precio de cada artículo sea de $50. Se pide obtener las tres ecuaciones (Ingreso, Costos y Utilidad).

Calculate the revenue, cost, and profit for a company producing and selling 75 fantasy items with a cost function of Ct = 30(x) + 1500 and a price of $50 per item. Understand linear functions in the context of business.

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