Geometry

We are given that a total length of 20 meters of wire is used to fence a circular sector, and we need to find the radius of the circle that maximizes the area of the sector.

Step 1: Formula for the perimeter of a sector
The perimeter of a circular sector consists of two radii and the arc length. If is the radius of the circle and is the central angle of the sector (in radians), then the perimeter of the sector is: We are given that the perimeter is 20 meters, so: This simplifies to:

Step 2: Formula for the area of the sector
The area of a sector is given by: We need to maximize this area with respect to .

Step 3: Express in terms of
From the perimeter equation, we can solve for :

Step 4: Substitute into the area formula
Substituting the expression for into the area formula, we get: Simplifying this:

Step 5: Maximize the area
To find the value of that maximizes the area, we take the derivative of with respect to : Setting to find the critical points:

Step 6: Verify that this is a maximum
The second derivative of is: Since the second derivative is negative, gives a maximum.

Answer: