Given Information:

• Initial temperature at 6:00 AM last Sunday: $45.1^\circ F$
• Rate of change of temperature is given by: $r(t) = 2.99 \cos \left( \frac{\pi t}{12} \right) + 0.036$ Where $t$ is the number of hours after 6:00 AM last Sunday.

We need to:

1. Find the temperature at 1:00 PM (7 hours after 6:00 AM).
2. Find the temperature at 4:00 PM last Friday (5 days or 120 hours after 6:00 AM last Sunday).

Solution:

Step 1: Temperature at 1:00 PM last Sunday

• The temperature increase from 6:00 AM to 1:00 PM is given as $11.76^\circ F$.
• Therefore, the temperature at 1:00 PM is: $45.1^\circ F + 11.76^\circ F = 56.86^\circ F$

This confirms the given temperature at 1:00 PM last Sunday is $56.86^\circ F$.

Step 2: Temperature at 4:00 PM last Friday (5 days later)

• We are asked to find the temperature 120 hours after 6:00 AM last Sunday.
• The rate function $r(t) = 2.99 \cos \left( \frac{\pi t}{12} \right) + 0.036$ repeats periodically with a 24-hour cycle. So, after 5 days, the temperature pattern will repeat similarly to the same time on Sunday.

At 4:00 PM last Friday, this would correspond to 10 hours after 6:00 AM, since 4:00 PM is 10 hours later than 6:00 AM. Hence, $t = 10$ for both Sunday and Friday.

Since we already know that the rate of temperature change function $r(t)$ repeats every 24 hours, we conclude that the temperature would be approximately the same as last Sunday at 4:00 PM.

The temperature at 4:00 PM is $54^\circ F$.

Final Answers:

1. The temperature at 1:00 PM last Sunday: $56.86^\circ F$.
2. The temperature at 4:00 PM last Friday: $54^\circ F$.

Let me know if you need further details or clarifications.

Here are 5 related questions for practice:

1. How would you find the temperature at any other specific time on Sunday using the rate function?
2. What is the temperature rate of change at 2:00 PM on Sunday?
3. How does the cosine function affect the periodicity of the temperature change?
4. What is the temperature 2 days after 6:00 AM Sunday, at noon?
5. How would the rate function change if the initial temperature was different?

Tip: When dealing with periodic functions like cosine in temperature models, it’s important to remember that the temperature follows a repeating cycle.