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The problem provides an exponential function to model the growth of bacteria on a kitchen counter. Here's the given function:

\[
A(t) = 10e^{0.06931t}
\]

Where:
- \(A(t)\) represents the number of bacteria at time \(t\) (in minutes),
- \(t\) is the time in minutes since the initial count of 10 bacteria,
- \(e\) is the base of the natural logarithm.

The problem asks how many bacteria will be on the counter 2.5 hours later. Let's break it down step by step:

### Step 1: Convert Hours to Minutes
Since the function uses time in minutes, we need to convert 2.5 hours to minutes:
\[
2.5 \, \text{hours} = 2.5 \times 60 = 150 \, \text{minutes}
\]

### Step 2: Plug Time into the Exponential Model
Now, substitute \(t = 150\) into the function:
\[
A(150) = 10e^{0.06931 \times 150}
\]
\[
A(150) = 10e^{10.3965}
\]

### Step 3: Calculate the Value of the Exponent
\[
e^{10.3965} \approx 32,557.53
\]

### Step 4: Final Calculation
Now multiply by 10:
\[
A(150) = 10 \times 32,557.53 = 325,575.3
\]

### Step 5: Round to the Nearest Whole Number
\[
A(150) \approx 325,575 \, \text{bacteria}
\]

Thus, after 2.5 hours, the predicted number of bacteria is approximately **325,575**.

---

Would you like more details or further clarifications?

### Related Questions:
1. What is the significance of the base \(e\) in exponential growth functions?
2. How would the result change if the doubling time were different, say 20 minutes instead of 10?
3. How can the constant 0.06931 be derived from the doubling time of 10 minutes?
4. How would the equation change if we started with a different initial amount of bacteria?
5. Can we apply this type of model to other real-life scenarios, such as population growth or radioactive decay?

### Tip:
In exponential growth problems, small changes in time or the growth rate can lead to significantly different results due to the nature of exponential functions.

Imagine that at a particular time, a kitchen counter has on it 10 bacteria and that this type of bacteria doubles in number every 10 minutes. The exponential function A(t) = 10e^0.06931t is a good model for predicting the number of bacteria A(t) in the population t minutes from some particular start time. How many bacteria could be predicted to be on the counter 2.5 hours later? Round to the nearest whole number.

Learn how to predict bacterial growth using the exponential function A(t) = 10e^0.06931t. This example shows how an initial population of 10 bacteria can grow exponentially over 2.5 hours. A detailed solution is provided, including the conversion from hours to minutes and calculating the bacteria population after 2.5 hours.

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