To find the dimensions of the rectangle with the maximum area that can be inscribed in a semicircle of radius $r = 10$ inches with two vertices on the diameter, we can use optimization methods.

Step-by-Step Solution:

1. Setup the Problem: Let the semicircle be centered at the origin on the coordinate plane, with the diameter along the $x$-axis, stretching from $-10$ to $10$.

2. Define the Rectangle's Vertices: Let the vertices of the rectangle lying on the diameter be at $(-x, 0)$ and $(x, 0)$. Then, the vertices on the curve will be at $(-x, y)$ and $(x, y)$.

3. Relationship Between $x$ and $y$: Since the top vertices lie on the semicircle, they must satisfy the equation of the semicircle: $x^2 + y^2 = r^2 = 100$ Solving for $y$, we get: $y = \sqrt{100 - x^2}$

4. Area of the Rectangle: The area $A$ of the rectangle is given by the width times the height. The width of the rectangle is $2x$ (distance from $-x$ to $x$), and the height is $y$. Thus: $A = 2x \cdot y = 2x \sqrt{100 - x^2}$

5. Maximize the Area Function: To find the maximum area, we take the derivative of $A$ with respect to $x$, set it equal to zero, and solve for $x$: $\frac{dA}{dx} = 2 \sqrt{100 - x^2} + 2x \cdot \frac{-x}{\sqrt{100 - x^2}} = 0$ Simplifying, $2 \sqrt{100 - x^2} = \frac{2x^2}{\sqrt{100 - x^2}}$ Cross-multiplying and solving for $x$, we get: $100 - x^2 = x^2$ $2x^2 = 100$ $x^2 = 50$ $x = \sqrt{50} = 5\sqrt{2} \approx 7.07 \text{ inches}$

6. Calculate $y$: Substitute $x = 5\sqrt{2}$ back into $y = \sqrt{100 - x^2}$: $y = \sqrt{100 - 50} = \sqrt{50} = 5\sqrt{2} \approx 7.07 \text{ inches}$

Conclusion:

The rectangle of maximum area that can be inscribed in the semicircle has:

• Length $2x = 10\sqrt{2} \approx 14.14$ inches
• Breadth $y = 5\sqrt{2} \approx 7.07$ inches

Would you like more details on any step, or do you have any questions?

---

Here are some related questions:

1. How do you calculate the area of other shapes inscribed in circles?
2. What if the rectangle were inscribed in a full circle instead of a semicircle?
3. Can this optimization method be applied to other shapes, like ellipses?
4. How does changing the radius of the semicircle affect the dimensions of the rectangle?
5. Are there applications of this problem in real-world design?

Tip: When maximizing or minimizing areas in similar problems, setting up an equation with variables in terms of a single parameter can simplify the solution.