Find the area of the largest rectangle that can be inscribed in a semicircle or radius r.
To find the area of the largest rectangle that can be inscribed in a semicircle of radius r, consider the following steps:
Step 1: Visualize the problem
Draw a semicircle with radius r. Now, imagine a rectangle inscribed inside the semicircle. The rectangle's height will be equal to the radius r, and the rectangle's width will be twice the radius r, forming a perfect rectangle.
Step 2: Determine the area of the rectangle
The area of a rectangle is equal to its length multiplied by its width. In this case, the length of the rectangle is r, and the width is 2r. Therefore, the area of the rectangle can be calculated as:
Area = length × width
= r × 2r
= 2r^2
Step 3: Simplify the area equation
To find the maximum area, differentiate the area equation with respect to r and set it equal to zero.
d(Area)/dr = 4r
Setting d(Area)/dr = 0:
4r = 0
r = 0
Since r cannot be zero, the condition does not satisfy the problem. So, there is no maximum area for a rectangle inscribed in a semicircle of radius r.
Conclusion:
The area of the largest rectangle that can be inscribed in a semicircle of radius r does not exist.