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The image shows a problem that involves an equation describing the temperature as a function of time. Let's break down the content.

The equation given in the problem is:

\[
T(t) = 60 + 10 \cdot \sin\left(\frac{\pi}{12} \cdot (t - 8)\right), \quad 0 \leq t \leq 24
\]

The context suggests that this function models the temperature at a certain location, where \( t \) is the time in hours after midnight.

The questions are:

a) ¿Cuál es la temperatura a las 5 a.m.?  
b) ¿A qué hora(s) se alcanza el máximo \( T(t) \)?  
c) Indica el máximo de \( T(t) \).  
d) Describe las características diarias, también se pide los tiempos a que ocurren.

Let's address each question.

### Part a) Calculating the Temperature at 5 a.m.
To find the temperature at 5 a.m., substitute \( t = 5 \) into the equation \( T(t) \).

\[
T(5) = 60 + 10 \cdot \sin\left(\frac{\pi}{12} \cdot (5 - 8)\right)
\]

Simplify the expression inside the sine function:

\[
T(5) = 60 + 10 \cdot \sin\left(\frac{\pi}{12} \cdot (-3)\right)
\]
\[
T(5) = 60 + 10 \cdot \sin\left(-\frac{3\pi}{12}\right)
\]
\[
T(5) = 60 + 10 \cdot \sin\left(-\frac{\pi}{4}\right)
\]

Since \(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\):

\[
T(5) = 60 + 10 \cdot \left(-\frac{\sqrt{2}}{2}\right)
\]
\[
T(5) = 60 - 5\sqrt{2}
\]

This is the temperature at 5 a.m. It can be approximately calculated as:

\[
T(5) \approx 60 - 7.07 = 52.93 \text{ degrees}
\]

### Part b) Finding the Time When \( T(t) \) is Maximum
The maximum of the sine function \(\sin(x)\) is 1. Therefore, the temperature \( T(t) \) is maximized when:

\[
\sin\left(\frac{\pi}{12} \cdot (t - 8)\right) = 1
\]

This happens when:

\[
\frac{\pi}{12} \cdot (t - 8) = \frac{\pi}{2}
\]

Solve for \( t \):

\[
t - 8 = 6 \quad \Rightarrow \quad t = 14
\]

So, the temperature reaches its maximum at 2 p.m. (14:00).

### Part c) The Maximum Value of \( T(t) \)
When \( \sin(x) = 1 \), the maximum value of \( T(t) \) is:

\[
T_{\text{max}} = 60 + 10 \cdot 1 = 70 \text{ degrees}
\]

### Part d) Daily Characteristics and Timing of Events
The temperature function is periodic with a period of 24 hours, meaning it repeats every day. The maximum temperature of 70 degrees occurs at 2 p.m., and the temperature oscillates between 50 degrees and 70 degrees. The lowest temperature occurs when the sine function reaches its minimum value of \(-1\), which happens 12 hours apart from the maximum, corresponding to \( t = 2 \) a.m.

Would you like more details or have any further questions?

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Here are five related questions to consider:

1. How would the temperature function change if the maximum temperature occurred at 1 p.m. instead of 2 p.m.?
2. What is the minimum temperature for this model, and when does it occur?
3. How would you modify the function to reflect a temperature range from 55 to 75 degrees instead of 50 to 70 degrees?
4. Can you determine the average temperature over a 24-hour period from this function?
5. How would the graph of this function look over a 24-hour period?

**Tip:** When dealing with sinusoidal functions in real-world contexts, understanding the amplitude and period is crucial for interpreting the model correctly.

Suponga que: T(t) = 60 + 10*sin((pi/12)*(t-8)), 0 <= t <= 24. Es un modelo matemático de la temperatura, en donde t es la hora después de la medianoche. a) ¿Cuál es la temperatura a las 5 a.m.? b) ¿A qué hora(s) se alcanza el máximo T(t)? c) Indica el máximo de T(t). d) Describe las características diarias, también se pide los tiempos a que ocurren.

Explore the sinusoidal temperature model equation T(t) = 60 + 10*sin((π/12)*(t-8)), solve for temperature at specific times, find the maximum temperature, and describe daily characteristics. Suitable for high school students in Grades 10-12.

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