To determine the average rate of change of the temperature for the specified time period, we need to follow these steps:
1. Identify the initial and final temperatures:
- The initial temperature at 7:30 am is [tex]\(-4^{\circ}F\)[/tex].
- The final temperature at 7:32 am is [tex]\(45^{\circ}F\)[/tex].
2. Calculate the change in temperature:
- The change in temperature is determined by subtracting the initial temperature from the final temperature:
[tex]\[
\Delta T = \text{final temperature} - \text{initial temperature} = 45^{\circ}F - (-4^{\circ}F)
\][/tex]
- Simplifying this:
[tex]\[
\Delta T = 45^{\circ}F + 4^{\circ}F = 49^{\circ}F
\][/tex]
3. Determine the time duration:
- Convert the times to minutes:
- 7:30 am is [tex]\(7 \times 60 + 30 = 450\)[/tex] minutes.
- 7:32 am is [tex]\(7 \times 60 + 32 = 452\)[/tex] minutes.
- The time duration is the difference between the two times:
[tex]\[
\Delta t = 452 \text{ minutes} - 450 \text{ minutes} = 2 \text{ minutes}
\][/tex]
4. Calculate the average rate of change of temperature:
- The average rate of change is the change in temperature divided by the time duration:
[tex]\[
\text{Average rate of change} = \frac{\Delta T}{\Delta t} = \frac{49^{\circ}F}{2 \text{ minutes}}
\][/tex]
- Simplifying this:
[tex]\[
\text{Average rate of change} = 24.5^{\circ}F \text{ per minute}
\][/tex]
Thus, the average rate of change of the temperature over this time period is [tex]\(24.5^{\circ}F\)[/tex] per minute.
