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The problem requires us to determine the lengths of the sides of the rectangle with maximum area that can be inscribed within an ellipse defined by the equation:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

### Approach:

1. **General Form of the Problem**:
   The vertices of the rectangle will lie on the ellipse, and we aim to maximize the area of this rectangle. The area of the rectangle is given by:

   \[
   A = 2x \cdot 2y = 4xy
   \]
   
   where \((x, y)\) are the coordinates of the vertices of the rectangle on the ellipse. Thus, the area we want to maximize is \(A = 4xy\).

2. **Relation from the Ellipse Equation**:
   From the ellipse equation, we have the constraint:

   \[
   \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
   \]
   
   Solving for \(y^2\) in terms of \(x\):

   \[
   y^2 = b^2 \left(1 - \frac{x^2}{a^2}\right)
   \]
   
   Taking the square root to express \(y\) as a function of \(x\):

   \[
   y = b \sqrt{1 - \frac{x^2}{a^2}}
   \]

3. **Maximizing the Area**:
   Now, substitute \(y = b \sqrt{1 - \frac{x^2}{a^2}}\) into the area expression \(A = 4xy\):

   \[
   A(x) = 4x \cdot b \sqrt{1 - \frac{x^2}{a^2}}
   \]

   To find the maximum area, we need to take the derivative of \(A(x)\) with respect to \(x\) and set it to 0.

   \[
   \frac{dA}{dx} = 4b \left( \sqrt{1 - \frac{x^2}{a^2}} - \frac{x^2}{a^2 \sqrt{1 - \frac{x^2}{a^2}}} \right) = 0
   \]

   Simplifying and solving for \(x\) will give us the critical points.

4. **Critical Points and Solution**:
   Solving this equation involves some algebra, but it results in \(x = \frac{a}{\sqrt{2}}\), and similarly, using the ellipse equation, \(y = \frac{b}{\sqrt{2}}\).

Thus, the dimensions of the rectangle that maximize the area are:

\[
\text{Length along the } x\text{-axis} = 2x = \frac{2a}{\sqrt{2}} = \sqrt{2}a
\]
\[
\text{Length along the } y\text{-axis} = 2y = \frac{2b}{\sqrt{2}} = \sqrt{2}b
\]

### Conclusion:
The lengths of the sides of the rectangle with maximum area inscribed in the ellipse are \( \sqrt{2}a \) and \( \sqrt{2}b \).

Would you like me to go into more detail about the derivative steps or the algebra? Also, here are five related questions to expand your understanding:

1. How would the solution change if the ellipse were rotated?
2. What if the rectangle were constrained to have a specific aspect ratio?
3. How do we maximize the area of a triangle inscribed in an ellipse?
4. Can the same method be applied to other shapes inscribed in ellipses, like rhombuses or trapezoids?
5. How does the maximum area change if we consider an ellipse with \( a = b \) (a circle)?

**Tip**: Optimization problems often involve taking derivatives and applying constraints using Lagrange multipliers or substitution.

Determine the lengths of the sides of a rectangle with maximum area that can be inscribed within the ellipse x^2/a^2 + y^2/b^2 = 1.

Learn how to determine the lengths of the sides of a rectangle with maximum area inscribed in an ellipse. This detailed solution involves optimization techniques, calculus, and ellipse geometry. Ideal for undergraduate students studying calculus.

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