Find the largest possible rectangular area you can enclose assuming you have 128 meters of fencing.
1024
The largest area is a square.
To find the largest possible rectangular area with a given amount of fencing, we need to determine the dimensions of the rectangle that maximize the area.
Let's assume the length of the rectangle is L meters and the width is W meters. From the given information, we know that the perimeter of the rectangle is 128 meters.
The perimeter of a rectangle is given by the formula:
Perimeter = 2(L + W)
Since we are given that the perimeter is 128 meters, we can express this mathematically as:
2(L + W) = 128
We can simplify this equation by dividing both sides by 2, which gives us:
L + W = 64
To find the largest possible area, we need to maximize the product of the length and width of the rectangle, which is given by the formula:
Area = L * W
To solve this problem, we can use the method of substitution. We can solve the equation L + W = 64 for one variable, and then substitute that into the area formula.
Let's solve for L in terms of W:
L = 64 - W
Substituting this into the area formula, we get:
Area = (64 - W) * W
To find the maximum area, we need to find the value of W that maximizes this expression. One way to do this is to plot a graph of the area as a function of W and find the maximum point.
However, an alternative method is to recognize that the area of a rectangle is maximized when its length and width are equal, i.e., L = W. So, let's set L = W and find the values for L and W.
L + W = 64
W + W = 64
2W = 64
W = 32
Therefore, the width of the rectangle is 32 meters.
Substituting this value back into L = 64 - W, we get:
L = 64 - 32
L = 32 meters
So, the largest possible rectangular area that can be enclosed with 128 meters of fencing is:
Area = L * W = 32 * 32 = 1024 square meters.
To find the largest possible rectangular area that can be enclosed with a specific amount of fencing, we can use the concept of optimization. In this case, we want to optimize the area of the rectangle given a fixed amount of fencing, which is 128 meters.
Let's assume the length of the rectangle is L meters and the width is W meters.
To maximize the area, we need to come up with an equation that relates the perimeter (fencing) with the area of the rectangle.
The perimeter of a rectangle is given by the formula:
Perimeter = 2L + 2W
Given that the perimeter is 128 meters, we can rewrite the formula as:
2L + 2W = 128
We can simplify the equation by dividing both sides by 2:
L + W = 64
Now let's solve for one of the variables in terms of the other. For example, we can solve for L:
L = 64 - W
The area of a rectangle is given by the formula:
Area = Length * Width
Substituting the expression for L:
Area = (64 - W) * W
This equation represents the area of the rectangle as a function of its width.
To find the largest possible area, we need to find the maximum value of this function. We can do that by taking the derivative of the function with respect to W, setting it equal to zero, and solving for W.
Let's differentiate the function:
d(Area)/dW = 64 - 2W
Setting the derivative equal to zero:
64 - 2W = 0
Solving for W:
2W = 64
W = 32
Now that we have the value for W, we can substitute it back into the equation we obtained earlier to find the corresponding value for L:
L = 64 - W = 64 - 32 = 32
So, the largest possible rectangular area that can be enclosed with 128 meters of fencing is obtained when the rectangle has dimensions of 32 meters (length) by 32 meters (width), resulting in an area of 32 * 32 = 1024 square meters.