https://r2.erweima.ai/mathbot/image/17264389429787b7mg3ep.jpg

Let's break down the problem from the image step by step.

We are given:

1. The company's weekly profits are decreasing exponentially, and they can be modeled by the equation:

\[
U(t) = 12,750,000 \cdot e^{kt}
\]

where:
- \( U(t) \) is the weekly profit in week \( t \).
- \( t \) is the number of weeks, with \( t = 0 \) corresponding to the last week of December 2023.
- \( k \) is a constant to be determined.

### Part a:

We need to calculate the constant \( k \), given that the company's profits in the 8th week (i.e., \( t = -8 \)) are 55% of the profits in the last week of December 2023 (i.e., at \( t = 0 \)).

This gives us the equation:
\[
U(-8) = 0.55 \cdot U(0)
\]

From the model, we know:
\[
U(t) = 12,750,000 \cdot e^{kt}
\]
Thus,
\[
U(0) = 12,750,000 \cdot e^{0} = 12,750,000
\]
and
\[
U(-8) = 12,750,000 \cdot e^{k(-8)} = 0.55 \cdot 12,750,000
\]

We can now solve for \( k \).

### Part b:

The company will stop operations when its weekly profits fall below 9% of the profits in the last week of December 2023. We need to find the time \( t \) when:
\[
U(t) < 0.09 \cdot U(0)
\]

This gives the inequality:
\[
12,750,000 \cdot e^{kt} < 0.09 \cdot 12,750,000
\]

We can solve for \( t \) once we know \( k \).

---

I'll start by solving for \( k \) in part a, and then proceed to part b.The constant \( k \) for the exponential model is approximately \( 0.07473 \).

Now, I'll proceed with part b to determine the week when the company's weekly profits fall below 9% of the profits from the last week of December 2023. We need to solve for \( t \) when:

\[
U(t) = 12,750,000 \cdot e^{kt} < 0.09 \cdot 12,750,000
\]

Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, the next step for part b is to solve the inequality:

\[
12,750,000 \cdot e^{0.07473t} < 0.09 \cdot 12,750,000
\]

This will give the value of \( t \) (the number of weeks before the company stops its operations). You can proceed by solving it manually or feel free to ask for further assistance.

Let me know if you need any clarification on this! Here are some additional related questions to explore:

1. How would you interpret the exponential decay of profits over time?
2. What real-world factors could contribute to an exponential decrease in profits?
3. How would changing the value of \( k \) affect the profit model?
4. What other methods could be used to predict a company's financial health besides exponential models?
5. How might a business adjust its operations in response to such a profit model?

**Tip:** When working with exponential models, always pay attention to the base of the exponential function, as it determines the rate of growth or decay.

Las utilidades semanales de una empresa de servicios han disminuido de manera exponencial desde la primera semana de enero de 2021. Se nos da la ecuación U(t) = 12,750,000 * e^(kt), donde t está medido en semanas (t=0 corresponde a la última semana de diciembre de 2023). Pregunta 4.2: a) Calcule la constante k del modelo exponencial, si las utilidades de la empresa en la octava semana son el 55% de las utilidades obtenidas por la empresa la última semana de diciembre de 2023. b) La empresa estima que debe finalizar sus operaciones si las utilidades semanales son menores al 9% de las utilidades obtenidas en la última semana de diciembre de 2023. Utilice el ítem anterior para determinar en qué semana la empresa paralizará sus operaciones.

Learn how to calculate the constant k in an exponential decay model for a service company's weekly profits and determine the time at which the company will cease operations based on profit thresholds. The problem involves exponential functions and percentage-based calculations, suitable for advanced high school or undergraduate students.

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