Let's break down the questions one by one based on the table provided in the image.

Given Table of Temperatures:

• Monday:
  • 8:00 AM: 100.5°F
  • 8:00 PM: 102.3°F
• Tuesday:
  • 8:00 AM: 103.1°F
  • 8:00 PM: 100.7°F
• Wednesday:
  • 8:00 AM: 99.9°F
  • 8:00 PM: 98.6°F

Question (a):

What was the average rate of change in Lisa’s temperature from 8:00 AM on Monday to 8:00 PM on Monday?

To calculate the average rate of change of Lisa's temperature, we use the formula:

$\text{Average Rate of Change} = \frac{\text{Change in Temperature}}{\text{Change in Time}}$

For Monday (from 8:00 AM to 8:00 PM):

• Temperature at 8:00 AM = 100.5°F
• Temperature at 8:00 PM = 102.3°F
• Time difference: 12 hours

$\text{Average Rate of Change} = \frac{102.3 - 100.5}{12} = \frac{1.8}{12} = 0.15 \, \text{°F per hour}$

Answer to (a): The average rate of change was 0.15°F per hour.

Question (b):

What was the average rate of change in Lisa’s temperature from 8:00 AM on Tuesday to 8:00 PM on Wednesday? Is your answer reasonable? What does the sign of the rate mean?

For Tuesday (8:00 AM) to Wednesday (8:00 PM):

• Temperature at 8:00 AM Tuesday = 103.1°F
• Temperature at 8:00 PM Wednesday = 98.6°F
• Time difference: 36 hours (8:00 AM Tuesday to 8:00 PM Wednesday)

$\text{Average Rate of Change} = \frac{98.6 - 103.1}{36} = \frac{-4.5}{36} = -0.125 \, \text{°F per hour}$

The negative sign indicates that Lisa's temperature was decreasing over this period.

Answer to (b): The average rate of change was $-0.125 \, \text{°F per hour}$. The negative sign indicates that her temperature was dropping. This is reasonable because the data shows a decreasing trend in temperature over this period.

Question (c):

During which 12-hour period was the average rate of change in Lisa’s temperature the greatest?

We will calculate the average rate of change for each 12-hour period and compare.

1. Monday (8:00 AM to 8:00 PM):

   • Already calculated: 0.15°F per hour

2. Tuesday (8:00 AM to 8:00 PM):

   • Temperature at 8:00 AM = 103.1°F
   • Temperature at 8:00 PM = 100.7°F
   • Time difference = 12 hours

$\text{Average Rate of Change} = \frac{100.7 - 103.1}{12} = \frac{-2.4}{12} = -0.2 \, \text{°F per hour}$

3. Wednesday (8:00 AM to 8:00 PM):
   • Temperature at 8:00 AM = 99.9°F
   • Temperature at 8:00 PM = 98.6°F
   • Time difference = 12 hours

$\text{Average Rate of Change} = \frac{98.6 - 99.9}{12} = \frac{-1.3}{12} = -0.1083 \, \text{°F per hour}$

Conclusion for (c): The greatest average rate of change was on Tuesday (8:00 AM to 8:00 PM), where the rate was $-0.2°F per hour$.

Would you like more details or have any further questions?

Here are 5 related questions you could explore:

1. What would the total temperature change be from Monday 8:00 AM to Wednesday 8:00 PM?
2. How can we interpret the rates of change in terms of Lisa's health?
3. What would the average rate of change be over the entire 3-day period?
4. How would the average rate of change look if we only consider day-to-day differences?
5. Can we model the temperature changes using a linear equation?

Tip: When calculating rates of change, a positive rate indicates an increase, while a negative rate indicates a decrease over the interval.