Final answer:
To maximize area with 52 feet of fencing, the enclosure should be a square with both the width and length being 13 feet, which results in a maximum area of 169 square feet.
Step-by-step explanation:
To solve this problem, we need to maximize the enclosed area with the given amount of fencing.
Including the gate, we have a total perimeter of 52 feet of fencing.
We will assume the enclosure is a rectangle, as a rectangle with given perimeter has the maximum area when it is a square.
Let width be w and length be l. Including the gate, the perimeter is given by:
2w + 2l = 52 feet.
If we solve the perimeter equation for l, we get:
l = 26 - w
The area( A ) of the rectangle is:
A = w × l
= w × (26 - w)
= 26w - w^2
This is a quadratic equation and opens downward, meaning it has a maximum value.
The vertex of this parabola, which gives the dimensions for the maximum area, occurs at w = b/2a when the general form is ax^2 + bx + c.
In this case:
w = 26/2(1)
= 13 feet
Therefore, the dimensions that create the maximum area are when the width and length are both 13 feet, giving an area of 169 square feet, which is a square.