A rectangular garden is to be fenced by a farmer. One side of the garden is next to the wall of the house and need not to be fenced. Find the dimensions of the largest garden that can be enclosed with 100m of fencing.

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Find the dimension of the largest garden that can be enclosed with 100m of fencing

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25m by 50m

To find the dimensions of the largest garden that can be enclosed with 100m of fencing, we can use the concept of optimization.

Let's assume that the length of the garden is L and the width is W. Since only three sides of the garden need to be fenced, we can derive the equation for the perimeter of the garden as follows:

Perimeter = Length + 2 * Width = L + 2W

According to the question, the total fencing available is 100m, so we can form an equation as follows:

L + 2W = 100

Now, we need to express L in terms of W to create an equation with a single variable. We can rearrange the equation above to solve for L:

L = 100 - 2W

To maximize the area of the garden, we need to find the dimensions that will give us the maximum area. The area of a rectangle is given by the equation:

Area = Length * Width = L * W

Substituting L from the equation above, we have:

Area = (100 - 2W) * W = 100W - 2W^2

Now, we have expressed the area of the garden as a function of a single variable W. To find the maximum area, we can take the derivative of the area equation with respect to W and set it equal to zero. Let's do that:

d(Area)/dW = 100 - 4W

Setting the derivative equal to zero:

100 - 4W = 0

Solving for W:

4W = 100
W = 100/4
W = 25

Now that we have W, we can substitute it back into the perimeter equation to find L:

L = 100 - 2W
L = 100 - 2(25)
L = 100 - 50
L = 50

Therefore, the dimensions of the largest garden that can be enclosed with 100m of fencing are 50m by 25m.

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