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Let's solve these two problems based on the data provided in the table:

### Given Data
- Time \( t \) (in minutes): \( 0, 20, 30, 50, 60 \)
- Temperature \( R(t) \) (in \( ^\circ \text{C} \)): \( 184.3, 140.3, 109.3, 51.3, 28.3 \)

### (a) Average Rate of Change from 0 to 30 minutes
To find the average rate of change, we use the formula:
\[
\text{Average Rate of Change} = \frac{R(t_2) - R(t_1)}{t_2 - t_1}
\]
For \( t_1 = 0 \) minutes and \( t_2 = 30 \) minutes:
- \( R(0) = 184.3 \)
- \( R(30) = 109.3 \)

Plugging in these values:
\[
\text{Average Rate of Change} = \frac{109.3 - 184.3}{30 - 0} = \frac{-75}{30} = -2.5 \, ^\circ\text{C per minute}
\]

So, the average rate of change from 0 to 30 minutes is **-2.5 °C per minute**.

### (b) Average Rate of Change from 50 to 60 minutes
For \( t_1 = 50 \) minutes and \( t_2 = 60 \) minutes:
- \( R(50) = 51.3 \)
- \( R(60) = 28.3 \)

Plugging in these values:
\[
\text{Average Rate of Change} = \frac{28.3 - 51.3}{60 - 50} = \frac{-23}{10} = -2.3 \, ^\circ\text{C per minute}
\]

So, the average rate of change from 50 to 60 minutes is **-2.3 °C per minute**.

Would you like further details or have any questions?

Here are some related questions you might consider:
1. What would be the rate of change from 20 to 50 minutes?
2. How would the results change if we extended the observation period to 90 minutes?
3. How can we determine the instantaneous rate of change at a specific time?
4. What would be the interpretation if the rate of change was positive?
5. How does the rate of change help in predicting the cooling process?

**Tip:** The average rate of change over a time interval gives a useful snapshot of the trend, but the instantaneous rate of change provides a more precise measurement at a specific moment.

Eric is cooking a roast. The table gives the temperature R(t) of the roast (in degrees Celsius) at a few times t (in minutes) after he removed it from the oven. The task is to calculate the average rate of change of the temperature over specific time intervals.

This problem involves calculating the average rate of change in temperature of a roast using a table of data over specific time intervals. The task applies the formula for average rate of change, demonstrating algebraic concepts suitable for students in Grades 9-10.

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