To find the area of the largest rectangle that can be inscribed in a semicircle of radius $r = 5$ cm, we can use optimization techniques.

Step 1: Set Up the Problem

1. Geometry: Consider a semicircle centered at the origin with radius 5 cm. The equation of the semicircle is: $x^2 + y^2 = r^2 \quad \text{(for } y \geq 0\text{)}$ Therefore, the semicircle can be expressed as: $y = \sqrt{25 - x^2}$

2. Rectangle Dimensions: Let the rectangle have width $2x$ and height $y$. The area $A$ of the rectangle is given by: $A = \text{width} \times \text{height} = 2x \cdot y = 2x \cdot \sqrt{25 - x^2}$

Step 2: Optimize the Area Function

To maximize the area, we need to find the derivative of $A$ and set it to zero.

1. Express Area: $A(x) = 2x \sqrt{25 - x^2}$

2. Differentiate: Using the product rule: $A'(x) = 2 \sqrt{25 - x^2} + 2x \cdot \frac{-x}{\sqrt{25 - x^2}}$ Simplifying, we get: $A'(x) = 2 \sqrt{25 - x^2} - \frac{2x^2}{\sqrt{25 - x^2}}$ Setting $A'(x) = 0$: $2 \sqrt{25 - x^2} = \frac{2x^2}{\sqrt{25 - x^2}}$ Cross-multiplying: $2(25 - x^2) = 2x^2 \implies 25 - x^2 = x^2 \implies 25 = 2x^2 \implies x^2 = 12.5 \implies x = \sqrt{12.5} = \frac{5\sqrt{2}}{2}$

3. Find $y$: Substitute $x$ back into the semicircle equation: $y = \sqrt{25 - x^2} = \sqrt{25 - 12.5} = \sqrt{12.5} = \frac{5\sqrt{2}}{2}$

Step 3: Calculate Maximum Area

Now we can calculate the maximum area: $A = 2x \cdot y = 2 \left( \frac{5\sqrt{2}}{2} \right) \left( \frac{5\sqrt{2}}{2} \right) = 2 \cdot \frac{25 \cdot 2}{4} = 25$

Conclusion

The area of the largest rectangle that can be inscribed in a semicircle of radius 5 cm is $\boxed{25 \text{ cm}^2}$.

Would you like more details or have any questions? Here are some related questions to consider:

1. What are other methods to find the maximum area of shapes inscribed in curves?
2. How does the area change if the radius of the semicircle is different?
3. Can similar optimization techniques be applied to different shapes?
4. What are real-world applications of inscribed shapes in geometry?
5. How does this problem relate to calculus concepts like derivatives and critical points?

Tip: When optimizing areas, always consider the constraints of the geometric shape involved.