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The most extreme change in temperature ever recorded was from [tex]-4^{\circ} F[/tex] to [tex]45^{\circ} F[/tex] between 7:30 am and 7:32 am on January 22, 1943, at Spearfish, South Dakota. What was the average rate of change of the temperature for this time period?



Answer :

GinnyAnswer
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.

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