Susan has 50 ft. of fencing in her backyard. What is the largest area garden Susan can create, show the gardens dimensions and explain how you know its the largest area

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Considering all rectangles with the same perimeter, the square encloses the greatest area.

Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding a to one side and subtracting a from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.

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To determine the largest area garden Susan can create, we need to find the dimensions that maximize the area while using the given 50 ft. of fencing.

Let's consider a rectangular garden since it typically maximizes the area for a given perimeter. Let's denote the length of the garden as L and the width as W.

We know that the perimeter, which is the total length of the fencing, is given by the equation:

Perimeter = 2(L + W)

In this case, the perimeter is 50 ft., so we can write the equation as:

50 = 2(L + W)

Now, we need to solve for either L or W in terms of the other variable. Let's solve for W:

50 = 2L + 2W
50 - 2L = 2W
25 - L = W

Now, we have an expression for the width of the garden (W) in terms of the length of the garden (L). We can substitute this expression into the formula for the area of a rectangle:

Area = Length × Width
Area = L × (25 - L)
Area = 25L - L^2

To find the largest possible area, we need to find the maximum value of this quadratic function. The maximum occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

In this case, a = -1 and b = 25. Substituting these values, we have:

L = -25 / (2 * -1)
L = 25 / 2

Since the length cannot be negative, we discard the negative value. Therefore, L = 25 / 2 = 12.5 ft.

Now, let's find the corresponding width using the earlier expression we derived:

W = 25 - L
W = 25 - 12.5
W = 12.5 ft.

So, the dimensions of the largest area garden Susan can create using 50 ft. of fencing are 12.5 ft. (length) by 12.5 ft. (width).

We know this is the largest area because we have mathematically determined the maximum value for the area function. By substituting L = 12.5 ft. and W = 12.5 ft. into the area equation (Area = 25L - L^2), we can calculate the actual largest area.