Alright, let's solve the problem step-by-step. The task requires us to find the average rate of change in temperature between 5 A.M. and 2 P.M., and then use this average rate to estimate the temperature at 3 P.M.
### Step 1: Identify Given Information
- Temperature at 5 A.M. (temp_5am): 68°F
- Temperature at 2 P.M. (temp_2pm): 100°F
- Time at 5 A.M. (time_5am): 5 hours (in 24-hour format, it's 5)
- Time at 2 P.M. (time_2pm): 14 hours (in 24-hour format, 2 P.M. is 14)
### Step 2: Calculate the Average Rate of Change
The average rate of change, often referred to as the slope in mathematical terms, can be calculated using the following formula:
[tex]\[ \text{Slope} = \frac{\text{Change in Temperature}}{\text{Change in Time}} = \frac{\Delta T}{\Delta t} \][/tex]
Using the given temperatures and times:
[tex]\[ \Delta T = \text{temp}_{2\text{pm}} - \text{temp}_{5\text{am}} = 100°F - 68°F = 32°F \][/tex]
[tex]\[ \Delta t = \text{time}_{2\text{pm}} - \text{time}_{5\text{am}} = 14\,\text{hours} - 5\,\text{hours} = 9\,\text{hours} \][/tex]
So the average rate of change (slope) will be:
[tex]\[ \text{Slope} = \frac{32°F}{9\,\text{hours}} \approx 3.5556\,°F \text{ per hour} \][/tex]
### Step 3: Estimate the Temperature at 3 P.M.
To determine the temperature at 3 P.M. (which is 15 hours in 24-hour format), we'll use the average rate of change. Here's how:
1. Find the difference in time between 2 P.M. (14 hours) and 3 P.M. (15 hours):
[tex]\[ \Delta t_{\text{new}} = 15\,\text{hours} - 14\,\text{hours} = 1\,\text{hour} \][/tex]
2. Multiply the average rate of change by this time difference to get the change in temperature:
[tex]\[ \Delta T_{\text{new}} = 3.5556\,°F/\text{hour} \times 1\,\text{hour} = 3.5556\,°F \][/tex]
3. Add this temperature change to the temperature at 2 P.M. to estimate the temperature at 3 P.M.:
[tex]\[ \text{temp}_{3\text{pm}} = \text{temp}_{2\text{pm}} + \Delta T_{\text{new}} \][/tex]
[tex]\[ \text{temp}_{3\text{pm}} = 100°F + 3.5556°F \approx 103.5556°F \][/tex]
### Final Answer
The average rate of change in temperature between 5 A.M. and 2 P.M. is approximately 3.5556°F per hour. Using this rate, the estimated temperature at 3 P.M. would be approximately 103.5556°F.
