The expression you provided is:

\[
P(3.5) = 149 \cdot e^{0.6931 \cdot 3.5}
\]

To solve this, we first calculate the exponent:

\[
0.6931 \cdot 3.5 = 2.42385
\]

Now, substitute this back into the equation:

\[
P(3.5) = 149 \cdot e^{2.42385}
\]

Next, we calculate \( e^{2.42385} \). Using a calculator or a scientific tool:

\[
e^{2.42385} \approx 11.246
\]

Finally, multiply by 149:

\[
P(3.5) \approx 149 \cdot 11.246 \approx 1673.554
\]

So:

\[
P(3.5) \approx 1673.55
\]

Would you like more details on any step of this process? Feel free to ask!

### Further questions:
1. What is the general formula for exponential growth models?
2. How does the base of the natural logarithm \( e \) relate to continuous growth?
3. What would happen if we changed the base of the exponent to something other than \( e \)?
4. How do you interpret the result \( P(3.5) \) in a real-world context, such as population growth or finance?
5. What is the significance of the constant \( 0.6931 \) in this equation?

### Tip:
When working with exponential functions, always make sure the exponent and base are clearly defined. The number \( e \) is crucial in many natural phenomena, especially those involving growth or decay.

Learn how to evaluate the exponential function P(3.5) = 149 ⋅ e^(0.6931 ⋅ 3.5) using step-by-step calculations. This problem covers key exponential concepts and natural exponent computations, providing a detailed solution for high school students in Grades 10-12.

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