A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 344 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?
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To find the largest area that can be enclosed, we need to determine the dimensions of the rectangular enclosure that maximizes the area. Let's assume that the length of the enclosure will be parallel to the city street, and the width will be perpendicular to it.
Let's denote the length of the enclosure as L (in feet) and the width as W (in feet). We are given that the developer has 344 feet of fencing. Since the side along the street is not fenced, we know that the perimeter of the rectangular enclosure is equal to three sides: L + 2W.
So, we can write the equation for the perimeter as:
L + 2W = 344
Now, we need to express the area, A, in terms of L and W. The area of a rectangle is given by multiplying its length and width:
A = L * W
We want to maximize the area A. We can rewrite the equation for the perimeter in terms of L:
L = 344 - 2W
Substituting this value of L into the equation for A, we get:
A = (344 - 2W) * W
To maximize the area, we can take the derivative of A with respect to W and set it equal to zero:
dA/dW = 0
Differentiating A with respect to W, we get:
dA/dW = 344 - 4W
Setting this derivative equal to zero and solving for W:
344 - 4W = 0
4W = 344
W = 86
Now that we have the value of W, we can substitute it into the equation for L:
L = 344 - 2W
L = 344 - 2 * 86
L = 172
Therefore, the dimensions of the enclosure that maximize the area are L = 172 feet and W = 86 feet.
Finally, we can calculate the maximum area by substituting these values into the equation for A:
A = L * W
A = 172 * 86
A ≈ 14,792 square feet
Thus, the largest area that can be enclosed is approximately 14,792 square feet.