A gardener has 160 meters of fencing to enclose a rectangular garden. What is the maximum area of the garden?

To determine the maximum area that can be enclosed by a rectangular garden with a given amount of fencing, we can use the relationship between perimeter and area. In this case, the gardener has 160 meters of fencing available, which corresponds to the perimeter of the rectangle.

The formula for the perimeter ( P ) of a rectangle is given by:

[ P = 2 \times (length + width) ]

Given that the perimeter is 160 meters, we can express the relationship between length ( l ) and width ( w ) as:

[ 160 = 2 \times (l + w) ]

This simplifies to:

[ l + w = 80 ]

For a given perimeter, the area ( A ) of the rectangle is maximized when the rectangle is a square. When the length and width are equal, ( l = w ):

Let ( l = w = x ):

[ l + w = x + x = 2x = 80 ]

[ x = 40 ]

So, both the length and width are 40 meters. The area ( A ) can be calculated as:

[ A = l \times w = 40 \times 40 = 160