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We are given the integral equation for the expected savings \( A(t) \) generated by a machine over \( t \) years:

\[
A(t) = \int_0^t (200x + 300) \, dx \, \text{(unidades monetarias)}
\]

The machine costs 5400 units of currency, and we are tasked with determining how many years it will take to recover this investment.

### Step 1: Solve the integral

\[
A(t) = \int_0^t (200x + 300) \, dx
\]

Let's integrate the function \( 200x + 300 \) with respect to \( x \):

\[
A(t) = \left[ 100x^2 + 300x \right]_0^t
\]

\[
A(t) = 100t^2 + 300t
\]

### Step 2: Set up the equation to recover the cost

To find how many years it will take to recover the investment, we need to set \( A(t) = 5400 \) (the machine's cost):

\[
100t^2 + 300t = 5400
\]

### Step 3: Solve the quadratic equation

Rearranging the equation:

\[
100t^2 + 300t - 5400 = 0
\]

Now, let's solve this quadratic equation. I'll calculate the solution for you.The solutions to the quadratic equation are \( t = -9 \) and \( t = 6 \). Since time cannot be negative, the valid solution is \( t = 6 \).

Thus, the company will recover its investment in 6 years.

Would you like further details or have any questions? Here are some related questions:

1. What happens to the savings function if the machine's efficiency increases?
2. How would the repayment period change if the machine's cost was higher?
3. What is the impact of changing the function to \( 250x + 400 \) instead of \( 200x + 300 \)?
4. How can we use this formula for different time horizons or varying interest rates?
5. What would be the impact if the company had ongoing maintenance costs?

**Tip:** Quadratic equations often have two solutions, but in practical problems, you should focus on the realistic one, such as positive time in this case.

¿En cuántos años se espera recuperar la inversión si el ahorro generado por la máquina está dado por la integral A(t) = ∫_0^t (200x + 300) dx, y la máquina costó 5400 unidades monetarias?

This problem explores the use of integral calculus to calculate the expected savings from a machine and determine how many years it will take to recover an investment of 5400 units of currency. The solution involves solving a quadratic equation after integrating the savings function A(t) = ∫_0^t (200x + 300) dx.

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