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
(a) What is the temperature of the refrigerator?

To find the temperature of the refrigerator, we need to determine the initial temperature of the roast before it was placed in the oven. This corresponds to the temperature at \( t = 0 \).

The temperature function is given by:

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

To find the temperature at \( t = 0 \), substitute \( t = 0 \) into the equation:

\[
R(0) = 350 - 313e^{-0.007(0)}
\]

Since \( e^{0} = 1 \), the equation simplifies to:

\[
R(0) = 350 - 313 \times 1 = 350 - 313 = 37 \text{ degrees Fahrenheit}
\]

So, the temperature of the refrigerator is **37 degrees Fahrenheit**.

Would you like more details or have any questions? 

Here are 5 related questions for further exploration:

1. What is the temperature of the roast after 30 minutes in the oven?
2. How long will it take for the roast to reach 100 degrees Fahrenheit?
3. What is the roast's temperature after 1 hour?
4. How does the value of the constant \(-0.007\) affect the rate at which the roast heats up?
5. What would be the final temperature of the roast as \( t \) approaches infinity?

**Tip:** Exponential decay functions like this one are often used to model cooling or heating processes because they accurately reflect how the rate of temperature change slows down as the object approaches its final temperature.

Learn how to find the initial temperature of a roast by using the exponential decay formula. This problem involves modeling temperature change using exponential functions, specifically the decay formula R(t) = 350 - 313e^{-0.007t}. Suitable for high school students studying calculus or physics.

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