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We are given a problem where a rectangle is inscribed in a semicircle of radius 10. The goal is to determine the function that models the area \( A \) of the rectangle in terms of its height \( h \).

### Step-by-step explanation:

1. **Geometry Setup:**
   - The semicircle has a radius of 10, so the equation of the semicircle is \( x^2 + h^2 = 10^2 \), which simplifies to:
     \[
     x^2 + h^2 = 100
     \]
     where \( h \) represents the height (the vertical side) of the rectangle, and \( x \) represents half the width of the rectangle.

2. **Width of the rectangle:**
   - Solving for \( x \), we get:
     \[
     x = \sqrt{100 - h^2}
     \]
     Since the full width of the rectangle is \( 2x \), the width is \( 2\sqrt{100 - h^2} \).

3. **Area of the rectangle:**
   - The area \( A(h) \) of the rectangle is given by the product of its width and height. Hence, the area function is:
     \[
     A(h) = 2\sqrt{100 - h^2} \cdot h
     \]
     Simplifying this expression gives:
     \[
     A(h) = 2h\sqrt{100 - h^2}
     \]

### Conclusion:
The correct expression for the area of the rectangle is given by option **D**:
\[
A(h) = 2h\sqrt{100 - h^2}, \quad 0 < h < 10
\]

Do you have any questions or would you like more details on any specific part?

Here are 5 related questions to explore:
1. How do we derive the equation for the semicircle based on the radius?
2. What happens to the area of the rectangle as \( h \) approaches 10?
3. Can you calculate the maximum possible area of the rectangle?
4. What are the dimensions of the rectangle when \( h = 5 \)?
5. How would the equation change if the semicircle had a different radius?

**Tip:** Always double-check the domain of the function, especially when dealing with geometric constraints like the radius of a semicircle.

Un rectángulo está inscrito en un semicírculo de radio 10. La función que modela el área A del rectángulo en términos de su altura h es:

Learn how to model the area of a rectangle inscribed in a semicircle of radius 10. This problem involves geometric and algebraic concepts, including the Pythagorean theorem and area calculations. Ideal for high school students in Grades 10-12.

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