The temperature in Middletown Park at​ 6:00 AM last Sunday was 45.1 degrees Fahrenheit.  The temperature was changing at a rate given by  r left parenthesis t right parenthesis equals 2.99 cosine left parenthesis StartFraction pi t Over 12 EndFraction right parenthesis plus 0.036

  where t is in hours after​ 6:00 AM last Sunday. 

ROUND ALL ANSWERS TO 2 DECIMAL PLACES.

At​ 10:00 AM last​ Sunday, the temperature in the park was increasing at a rate of 

about 
  
1.53 degrees per hour.

From​ 6:00 AM to​ 1:00 PM last​ Sunday, the temperature in the park increased

by 
  
11.76 degrees.

What was the temperature in the park at​ 1:00 PM last​ Sunday?   
  
56.86 degrees

What was the temperature in the park at​ 4:00 PM Last Friday​ (5 days​ later)?   
  
54 degrees

### 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.

The temperature in Middletown Park at​ 6:00 AM last Sunday was 45.1 degrees Fahrenheit. The temperature was changing at a rate given by r(t) = 2.99 cos(πt/12) + 0.036 where t is in hours after​ 6:00 AM last Sunday. At 10:00 AM last​ Sunday, the temperature in the park was increasing at a rate of about 1.53 degrees per hour. From​ 6:00 AM to​ 1:00 PM last​ Sunday, the temperature in the park increased by 11.76 degrees. What was the temperature in the park at​ 1:00 PM last​ Sunday? 56.86 degrees. What was the temperature in the park at​ 4:00 PM Last Friday​ (5 days​ later)? 54 degrees.

Solve a temperature change problem using a cosine function in Middletown Park. Learn how the temperature increased by 11.76 degrees between 6:00 AM and 1:00 PM last Sunday and determine the temperature at specific times. Covers key trigonometric concepts and periodic behavior of functions. Suitable for Grades 11-12.

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