A roast is taken from the refrigerator (where it had been for several days) and placed immediately in a preheated oven to cook. The temperature R = R(t) of the roast t minutes after being placed in the oven is given below.
R = 350 − 313e−0.007t degrees Fahrenheit.  By how much did the temperature of the roast increase from the first hour to 8 minutes after the first hour of cooking?

To solve this problem, we need to calculate the temperature of the roast at two specific times and then find the difference between these temperatures.

### Step 1: Determine the temperature after the first hour (60 minutes).
The temperature \( R(t) \) is given by the function:

\[
R(t) = 350 - 313e^{-0.007t}
\]

Substitute \( t = 60 \) minutes into the equation:

\[
R(60) = 350 - 313e^{-0.007 \times 60}
\]

### Step 2: Determine the temperature after 68 minutes (8 minutes after the first hour).
Substitute \( t = 68 \) minutes into the equation:

\[
R(68) = 350 - 313e^{-0.007 \times 68}
\]

### Step 3: Calculate the difference in temperature.
The difference in temperature from 60 minutes to 68 minutes is:

\[
\Delta R = R(68) - R(60)
\]

Let's calculate these values.

First, calculate \( R(60) \):

\[
R(60) = 350 - 313e^{-0.007 \times 60}
\]

\[
R(60) = 350 - 313e^{-0.42}
\]

Then, calculate \( R(68) \):

\[
R(68) = 350 - 313e^{-0.007 \times 68}
\]

\[
R(68) = 350 - 313e^{-0.476}
\]

Finally, subtract \( R(60) \) from \( R(68) \).

I'll compute these values for you.The temperature of the roast increased by approximately 11.2 degrees Fahrenheit from the first hour to 8 minutes after the first hour of cooking.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the rate constant \( k \) affect the temperature increase?
2. What would the temperature be after 2 hours of cooking?
3. How long would it take for the roast to reach a temperature of 300 degrees Fahrenheit?
4. What does the exponential decay term \( e^{-0.007t} \) represent in the context of this problem?
5. How does the initial temperature of the roast affect the overall cooking process?

**Tip:** Understanding exponential functions is crucial for analyzing processes like cooling and heating, where rates change over time.

Learn how to calculate the temperature increase of a roast over time using an exponential function model. This problem involves determining temperature changes from specific time intervals in a cooking process. Suitable for high school students and those interested in understanding exponential functions in real-world scenarios.

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