A potato is placed in a preheated oven to bake. Its temperature P = P(t) is given by the following.
P = 400 − 325e-t/50 What was the average rate of change per minute during the first 30 minutes? (Round your answer to two decimal places.)

The average rate of change of the temperature \( P(t) \) over a given time interval \([a, b]\) is found using the formula:

\[
\text{Average rate of change} = \frac{P(b) - P(a)}{b - a}
\]

For this problem, the function describing the temperature is:

\[
P(t) = 400 - 325e^{-t/50}
\]

We are asked to find the average rate of change during the first 30 minutes, so the interval is from \( t = 0 \) to \( t = 30 \). Let's calculate:

1. Find \( P(0) \):
\[
P(0) = 400 - 325e^{0} = 400 - 325 = 75
\]

2. Find \( P(30) \):
\[
P(30) = 400 - 325e^{-30/50} = 400 - 325e^{-0.6}
\]
We can compute \( e^{-0.6} \) to find the exact value of \( P(30) \).

Now, I'll calculate these values.The temperature of the potato at \( t = 0 \) minutes is \( P(0) = 75 \)°F, and at \( t = 30 \) minutes, it is approximately \( P(30) = 221.64 \)°F. 

The average rate of change of the temperature during the first 30 minutes is approximately \( 4.89 \)°F per minute.

Would you like more details, or do you have any further questions?

Here are 5 related questions:
1. How can the derivative of \( P(t) \) help interpret the rate of temperature change at any time?
2. What would the temperature be after 60 minutes?
3. How does the exponential term \( e^{-t/50} \) affect the long-term behavior of the temperature?
4. How would the average rate of change differ if the time interval were extended to 60 minutes?
5. What is the temperature approaching as time goes to infinity?

**Tip:** When dealing with exponential decay, as time increases, the effect of the exponential term diminishes, leading the function towards a constant value.

A potato is placed in a preheated oven to bake. Its temperature P = P(t) is given by the following. P = 400 − 325e^(-t/50). What was the average rate of change per minute during the first 30 minutes? (Round your answer to two decimal places.)

Learn how to calculate the average rate of change of a potato's temperature during the first 30 minutes of baking using the exponential decay function P(t) = 400 - 325e^(-t/50). Step-by-step solution suitable for calculus students in Grades 11-12 or college-level courses.

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