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The problem involves the resale value of a radiographic machine that decreases exponentially over time according to the formula:

\[
V = 5500e^{-0.05t}
\]

where:
- \( V \) is the value of the equipment in soles.
- \( t \) is the number of years since the equipment was purchased.

Let's address each question:

### a) ¿Cuáles son las variables en estudio?
The variables in the study are:
- \( t \): the time in years since the purchase of the equipment.
- \( V \): the resale value of the equipment in soles.

### b) Represente numéricamente el tiempo (los años) en el momento en el que Manuel realizó la compra del equipo radiográfico.
At the time of purchase, the time \( t \) is:
- \( t = 0 \) years.

### c) De acuerdo con el comportamiento de la reventa del equipo radiográfico, ¿cuánto fue el valor de este equipo cuando fue nuevo?
To find the initial value when the equipment was new, substitute \( t = 0 \) into the equation:

\[
V = 5500e^{-0.05 \times 0} = 5500 \times 1 = 5500 \text{ soles}
\]

So, the initial value of the equipment was 5500 soles.

### d) ¿Después de cuántos años el valor de reventa será de 2500 soles? Aproxime a 2 decimales.
We need to solve the equation \( V = 2500 \) for \( t \):

\[
2500 = 5500e^{-0.05t}
\]

Dividing both sides by 5500:

\[
\frac{2500}{5500} = e^{-0.05t}
\]

\[
\frac{5}{11} = e^{-0.05t}
\]

Taking the natural logarithm on both sides:

\[
\ln\left(\frac{5}{11}\right) = -0.05t
\]

\[
t = \frac{\ln\left(\frac{5}{11}\right)}{-0.05}
\]

Let's calculate this value:

The value of the equipment will be 2500 soles approximately after **15.77 years**.

### Follow-up Questions:
1. What happens to the value of the equipment after 20 years?
2. How does the depreciation rate affect the value of the equipment?
3. What would the value be after 10 years?
4. How would the equation change if the initial value was 6000 soles?
5. Can this model be used for other types of equipment?

**Tip:** Exponential decay models like this are common for valuing assets that lose value over time, such as medical or technological equipment.

Manuel, a prestigious doctor in Lima, needs to change some medical equipment. He travels to downtown Lima to sell one of his medical devices. He is willing to sell his radiographic equipment, which follows the equation V = 5500e^{-0.05t}, where t represents the years since the purchase. The following questions are asked:

a) What are the variables under study? (2 pts.)

b) Numerically represent the time (years) at which Manuel purchased the radiographic equipment. (2 pts.)

c) Based on the behavior of the resale value of the radiographic equipment, what was the value of this equipment when it was new? (2 pts.)

d) After how many years will the resale value be 2500 soles? Approximate to 2 decimals. (6 pts.)

Solve an exponential decay problem involving the resale value of radiographic equipment using the formula V = 5500e^{-0.05t}. This problem covers key mathematical concepts, including determining variables, initial value, and solving for a specific resale value over time. Suitable for undergraduate or advanced high school students.

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