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We are tasked with solving the equation:

\[
98e^{\frac{x}{2}} = 490
\]

### Step-by-step solution:

1. **Isolate the exponential term**:
   \[
   e^{\frac{x}{2}} = \frac{490}{98}
   \]
   Simplifying the right-hand side:
   \[
   e^{\frac{x}{2}} = 5
   \]

2. **Take the natural logarithm (ln) of both sides** to eliminate the exponential:
   \[
   \ln\left(e^{\frac{x}{2}}\right) = \ln(5)
   \]
   By logarithmic properties, \(\ln(e^y) = y\), so:
   \[
   \frac{x}{2} = \ln(5)
   \]

3. **Solve for \(x\)**:
   \[
   x = 2 \cdot \ln(5)
   \]
   Using a calculator to evaluate \(\ln(5)\):
   \[
   \ln(5) \approx 1.60944
   \]
   Therefore:
   \[
   x = 2 \cdot 1.60944 = 3.21888
   \]

### Final answer:
\[
x \approx 3.22
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the solution change if the equation were \(98e^{\frac{x}{2}} = 600\)?
2. What is the general method to solve equations involving exponential functions?
3. How does the base of the natural logarithm, \(e\), influence the solution process?
4. What is the difference between natural logarithms and logarithms with other bases?
5. How do you solve exponential equations when the exponent includes variables in more complex terms?

**Tip**: Remember, when solving exponential equations, using logarithms allows you to bring down exponents and solve for variables.

Solve for x, rounding to the nearest hundredth. 98e^(x/2) = 490

Learn how to solve the exponential equation 98e^(x/2) = 490 with a detailed step-by-step solution. This problem involves isolating the exponential, applying logarithms, and solving for x. Suitable for students in Grades 10-12.

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