A farmer wants to fence in part of her land so that her chickens will have their own little area. If she only has 28 feet of fence, what is the area of the largest pen that she can build in square feet?

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To find the area of the largest pen the farmer can build, we need to determine the shape of the pen that will maximize the area within the given 28 feet of fence.

Let's consider a rectangular pen, as it will likely provide the largest area.

A rectangular pen has two equal sides and two equal widths. Let's assume the sides of the pen have length 'x' feet and the widths have length 'y' feet.

The perimeter of the pen is the sum of all the sides: 2x (for the length) + 2y (for the width). We are given that the total fence length is 28 feet: 2x + 2y = 28.

To find the area, we'll use the formula: Area = Length * Width = x * y.

To maximize the area, we need to solve the perimeter equation for one of the variables and substitute it into the area equation.

Rearranging the perimeter equation, we have:
2x + 2y = 28
=> 2y = 28 - 2x
=> y = 14 - x/2

Substituting this value of y into the area equation:
Area = x * (14 - x/2)

Now, we have an equation to represent the area in terms of x. To find the maximum area, we can differentiate the area equation with respect to x, and set the derivative equal to zero.

d(Area)/dx = (14 - x/2) - x/2 = 14 - x

Setting the derivative equal to zero:
14 - x = 0
=> x = 14

Substituting this value of x back into the equation for y:
y = 14 - 14/2
=> y = 7

So, the dimensions of the pen that maximize its area are:
Length = 14 feet
Width = 7 feet

To find the area, multiply the length and width:
Area = 14 * 7
=> Area = 98 square feet

Therefore, the largest pen the farmer can build has an area of 98 square feet.