To solve this problem, we need to evaluate the function $f(x) = 300e^{(0.02x)}$ at $x = 48$, and then round the answer to three decimal places.

Steps to solve:

1. Substitute $x = 48$: $f(48) = 300e^{(0.02 \times 48)}$

2. Calculate the exponent: $0.02 \times 48 = 0.96$

   So, we now have: $f(48) = 300e^{0.96}$

3. Evaluate $e^{0.96}$ (using the value of $e \approx 2.718$): $e^{0.96} \approx 2.6117$

4. Multiply by 300: $f(48) \approx 300 \times 2.6117 = 783.51$

5. Round to three decimal places: $f(48) \approx 783.510$

Final Answer:

$f(48) = 783.510$

Would you like a more detailed explanation or have any questions?

Here are some related questions to deepen understanding:

1. What would $f(x)$ equal if $x = 50$ instead of 48?
2. How does changing the coefficient 300 affect the function's value?
3. What is the impact of changing the exponent base $e$ to a different value?
4. How can we interpret the significance of $e^{0.02x}$ in real-world applications?
5. What happens to $f(x)$ as $x$ approaches infinity?

Tip: Always double-check exponent calculations, as small errors in exponent values can significantly impact the result.