To solve the problem of finding the maximum area of a rectangle inscribed in a semicircle of radius (r), follow these steps:

Step 1: Model the Problem

Place the semicircle on the coordinate system with center at ((0,0)) and radius (r) (upper half: (y\geq0)). The equation of the semicircle is:
[x^2 + y^2 = r^2 \quad (y\geq0)]

A rectangle inscribed in the semicircle has vertices ((-x,0), (x,0), (x,y), (-x,y)). Its length is (2x) and height is (y). The area (A) is:
[A = 2x \cdot y]

Step 2: Express Area in Terms of One Variable

From the semicircle equation: (y = \sqrt{r^2 - x^2}). Substitute into the area formula:
[A(x) = 2x\sqrt{r^2 - x^2}]

Step 3: Maximize the Area

Using trigonometry for simplicity: let (x = r\cos\theta), (y = r\sin\theta). Then:
[A = 2(r\cos\theta)(r\sin\theta) = r^2 \cdot 2\sin\theta\cos\theta = r^2\sin(2\theta)]

The maximum value of (\sin(2\theta)) is (1) (when (\theta=45^\circ)). Thus:
[A_{\text{max}} = r^2 \cdot 1 = r^2]

Answer: (\boxed{r^2})

(Note: If the problem uses a different variable like (R), replace (r) with (R).)

(\boxed{r^2})

作者声明:本文包含人工智能生成内容。