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The problem asks to find the average rate of change of the temperature function \( H(t) \) between \( t = 0 \) and \( t = 20 \), where the temperature \( H(t) \) is measured in degrees Fahrenheit, and the corresponding values for \( t \) and \( H(t) \) are provided in the table.

### Table of values:
\[
\begin{array}{|c|c|}
\hline
t \ (\text{min}) & H(t) \ (\text{°F}) \\
\hline
0 & 170 \\
2 & 137 \\
5 & 107 \\
10 & 83 \\
13 & 77 \\
20 & 72 \\
\hline
\end{array}
\]

### Average Rate of Change Formula:
The average rate of change of a function between two points \( t_1 \) and \( t_2 \) is given by:
\[
\frac{H(t_2) - H(t_1)}{t_2 - t_1}
\]

Here, \( t_1 = 0 \), \( t_2 = 20 \), \( H(t_1) = H(0) = 170 \), and \( H(t_2) = H(20) = 72 \).

### Calculation:
\[
\frac{H(20) - H(0)}{20 - 0} = \frac{72 - 170}{20} = \frac{-98}{20} = -4.9 \ \text{°F/min}
\]

### Conclusion:
The average rate of change of the temperature between \( t = 0 \) and \( t = 20 \) is \( -4.9 \ \text{°F/min} \).

Would you like a deeper explanation or have any questions?

#### Related Questions:
1. What does a negative average rate of change represent in this context?
2. How does the rate of change compare between smaller intervals, such as \( t = 0 \) to \( t = 2 \)?
3. How can we approximate the instantaneous rate of change using this table?
4. What would happen to the rate of change if we extended the time interval beyond 20 minutes?
5. How is the average rate of change related to the slope of the line connecting two points on the graph of \( H(t) \)?

**Tip:** The average rate of change gives an overall sense of how a quantity changes over an interval, while the instantaneous rate of change focuses on behavior at specific points.

Find the average rate of change of the temperature of hot chocolate between t = 0 and t = 20. Use proper units in your answer.

This problem explores how to calculate the average rate of change of a temperature function H(t) over an interval from t = 0 to t = 20 minutes, where the temperature decreases from 170°F to 72°F. Learn how to apply the average rate of change formula for proper understanding.

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