Questions: At 12 pm, the temperature was 57 degrees F. By 5 pm, the thermometer read 36 degrees F. Assuming a constant change in temperature, calculate and interpret the average rate of change.

At 12 pm, the temperature was 57 degrees F. By 5 pm, the thermometer read 36 degrees F. Assuming a constant change in temperature, calculate and interpret the average rate of change.

Transcript text: At 12 pm , the temperature was $57^{\circ} \mathrm{F}$. By 5 pm , the thermometer read $36^{\circ} \mathrm{F}$. Assuming a constant change in temperature, calculate and interpret the average rate of change.

Solution

Solution Steps

To solve this problem, we need to calculate the average rate of change in temperature over the given time period. The average rate of change is found by taking the difference in temperature and dividing it by the difference in time.

Calculate the difference in temperature between 12 pm and 5 pm.
Calculate the difference in time between 12 pm and 5 pm.
Divide the difference in temperature by the difference in time to find the average rate of change.

Step 1: Calculate Temperature Difference

The temperature at 12 pm is $ T_{12} = 57^{\circ} \mathrm{F} $ and at 5 pm is $ T_{5} = 36^{\circ} \mathrm{F} $. The difference in temperature is calculated as: \[ \Delta T = T_{12} - T_{5} = 57 - 36 = 21^{\circ} \mathrm{F} \]

Step 2: Calculate Time Difference

The time from 12 pm to 5 pm is $ \Delta t = 5 - 12 = -7 $ hours. Since we are considering the time in a 12-hour format, we can express this as: \[ \Delta t = 5 \text{ hours} \]

Step 3: Calculate Average Rate of Change

The average rate of change in temperature over the time period is given by: \[ \text{Average Rate of Change} = \frac{\Delta T}{\Delta t} = \frac{21^{\circ} \mathrm{F}}{5 \text{ hours}} = 4.2^{\circ} \mathrm{F/hour} \]