We are given that a rectangle is inscribed in a circle of radius 5, and we need to express the area of the rectangle as a function of $x$.

Step 1: Equation of the Circle

The circle has a radius of 5, so its equation is: $x^2 + y^2 = 25$ The point $P(x, y)$ is on the circle and represents one of the vertices of the rectangle in the first quadrant.

Step 2: Relationship Between $x$ and $y$

From the equation of the circle, we can express $y$ in terms of $x$: $y = \sqrt{25 - x^2}$

Step 3: Dimensions of the Rectangle

Since the rectangle is symmetric with respect to both the x-axis and y-axis, its width is $2x$ (because the vertex at $P(x, y)$ is at $x$, and the other vertex is at $-x$) and its height is $2y$ (similarly, the height extends from $y$ to $-y$).

• Width = $2x$
• Height = $2y = 2\sqrt{25 - x^2}$

Step 4: Area of the Rectangle

The area $A(x)$ of the rectangle is the product of its width and height: $A(x) = 2x \cdot 2\sqrt{25 - x^2}$ Simplifying: $A(x) = 4x\sqrt{25 - x^2}$

Thus, the area of the rectangle as a function of $x$ is: $A(x) = 4x\sqrt{25 - x^2}$

Let me know if you'd like further clarification!

Here are 5 related questions to expand your understanding:

1. How does the area of the rectangle change as $x$ approaches 0?
2. What is the maximum possible area for the rectangle, and at what $x$ does it occur?
3. How would the equation change if the radius of the circle were different?
4. What are the coordinates of the vertex $P$ when the rectangle has its maximum area?
5. How does the symmetry of the problem help in simplifying the calculations?

Tip: To find the maximum area of the rectangle, you can take the derivative of $A(x)$ and find its critical points.