The problem asks us to analyze a rectangle inscribed in a semicircle with a radius of 20. We're provided with the equation for the area $A(x) = 2x \sqrt{400 - x^2}$, which gives the area of the rectangle in terms of $x$, the length of one side of the rectangle. We are also given the derivative $A'(x)$ and critical points for $A(x)$.

To solve this problem:

1. Critical Points: We need to find the dimensions of the rectangle at the critical point where $A'(x) = 0$, which is provided to be $x = 10\sqrt{2}$.

2. Maximum and Minimum Areas: To determine which rectangle corresponds to the maximum or minimum area, we examine the second derivative or analyze the sign change of $A'(x)$.

3. Dimensions: For $x = 10\sqrt{2}$, the dimensions of the rectangle are:

   • Width: $2x$
   • Height: $\sqrt{400 - x^2}$

We will now compute the dimensions at $x = 10\sqrt{2}$ and verify which option is correct.

Let me proceed with these calculations. sympy as sp

Define the symbol and the equation for A(x)

x = sp.symbols('x') A = 2 * x * sp.sqrt(400 - x**2)

Compute A'(x) to check the critical points

A_prime = sp.diff(A, x)

Define x = 10*sqrt(2) to calculate the width and height of the rectangle

critical_x = 10 * sp.sqrt(2)

Calculate the width (2x) and height (sqrt(400 - x^2)) at x = 10*sqrt(2)

width = 2 * critical_x height = sp.sqrt(400 - critical_x**2)

width, heightAt the critical point $x = 10\sqrt{2}$, the dimensions of the inscribed rectangle are:

• Width: $20\sqrt{2}$
• Height: $10\sqrt{2}$

Now, looking at the given options:

• Option A: The inscribed rectangle with maximum area has dimensions $10\sqrt{2}$ by $10\sqrt{2}$.
• Option B: The inscribed rectangle with minimum area has dimensions $10\sqrt{2}$ by $10\sqrt{2}$.
• Option C: The inscribed rectangle with maximum area has dimensions $20\sqrt{2}$ by $10\sqrt{2}$.
• Option D: The inscribed rectangle with minimum area has dimensions $20\sqrt{2}$ by $10\sqrt{2}$.

The correct answer is Option C, which corresponds to the maximum area.

Would you like further details or have any questions about the steps? Here are some related questions to explore further:

1. How do we confirm whether a critical point is a maximum or a minimum using the second derivative test?
2. How can we derive the general formula for the area of an inscribed rectangle in any semicircle?
3. What role does symmetry play in the dimensions of the rectangle?
4. What happens when the value of $x$ approaches 20 in this problem?
5. How do we analyze the behavior of $A(x)$ at the boundaries (i.e., when $x = 0$ or $x = 20$)?

Tip: When dealing with optimization problems like this, always start by finding the derivative, setting it equal to zero to find critical points, and then use the second derivative test or analyze the behavior of the function at the critical points.