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The problem is asking to find the exact height of the curve given by:

\[
y = \ln\left(1 - x^2\right)
\]

over the interval \( 0 \leq x \leq \frac{\sqrt{5}}{5} \).

### Steps to Find the Maximum Height:

1. **Understand the Function:**
   - The function \( y = \ln(1 - x^2) \) is defined as long as \( 1 - x^2 > 0 \), which means \( |x| < 1 \). This condition is satisfied within the given interval.

2. **Differentiate to Find Critical Points:**
   - To find the maximum or minimum values of the function, take the derivative with respect to \( x \) and set it to zero.
   \[
   \frac{dy}{dx} = \frac{d}{dx} \ln(1 - x^2) = \frac{1}{1 - x^2} \cdot (-2x) = \frac{-2x}{1 - x^2}
   \]
   - Set \( \frac{dy}{dx} = 0 \) to find critical points:
   \[
   \frac{-2x}{1 - x^2} = 0
   \]
   This equation is satisfied when \( x = 0 \).

3. **Evaluate the Function at the Critical Points and Endpoints:**
   - At \( x = 0 \):
   \[
   y = \ln(1 - 0^2) = \ln(1) = 0
   \]
   - At \( x = \frac{\sqrt{5}}{5} \):
   \[
   y = \ln\left(1 - \left(\frac{\sqrt{5}}{5}\right)^2\right) = \ln\left(1 - \frac{5}{25}\right) = \ln\left(\frac{20}{25}\right) = \ln\left(\frac{4}{5}\right)
   \]
   Since \( \ln\left(\frac{4}{5}\right) \) is negative, the maximum value occurs at \( x = 0 \).

### Conclusion:
The exact height of the curve is **0**.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What is the value of \( y \) at \( x = \frac{\sqrt{5}}{5} \), and what does this tell you about the curve's behavior?
2. How would the function behave if the interval were extended to include negative values of \( x \)?
3. What would happen to the maximum height if the domain of \( x \) were different, say \( 0 \leq x \leq 1 \)?
4. Can you derive a general method to find the maximum value of \( y = \ln(1 - kx^2) \) for any constant \( k \)?
5. How does the concavity of the function \( y = \ln(1 - x^2) \) help in identifying maximum points?

**Tip:** Always evaluate the function at the endpoints of the given interval as well as any critical points to find the maximum or minimum values.

Learn how to find the maximum height of the curve y = ln(1 - x^2) over the interval [0, sqrt(5)/5]. Detailed step-by-step solution including differentiation, critical points evaluation, and endpoint analysis. Ideal for calculus students seeking to understand optimization problems.

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