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Now, use your knowledge about slope and the slope formula to answer the following questions...
7. In the Mojave Desert in California, temperatures can drop quickly from day to night. Assuming the
temperature rises until at least 3 P.M. and given average temperatures of 68°F at 5 A.M and 100°F
at 2 P.M. to. Find the average rate of change and use it to determine the temperature at 3 P.M.

Sagot :

GinnyAnswer
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.
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