what is the largest are of rectangular area that can be enclosed in sixteen meter fence
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I am sure that I have answered this one recently
Let the width of the rectangle = w and the height = h
2w+2h=16 or
w=8-h
the area=wh or
area = (8-h)h = 8h-h^2 =a
there are a couple of ways to finding the maximum area. You could plot area against h and find where h is a maximum.
Alternatively the area is a maximum when da/dh=0
da/dh=8-2h, which is zero when h=4
i.e. the maximum area is when h=4 and w=4
so the largest area is 16 m^2
To find the largest area of a rectangular area that can be enclosed by a given fence length, we need to apply some mathematical reasoning:
Let's assume the length of the rectangle is L and the width is W.
The perimeter of a rectangle is given by the formula:
Perimeter = 2L + 2W
In this case, we know that the perimeter is 16 meters. So we can write the equation as:
2L + 2W = 16
Simplifying the equation further, we get:
L + W = 8
To maximize the area, we can use the fact that the area of a rectangle is given by the formula:
Area = Length × Width
We can rewrite the formula to express the width in terms of the length:
W = 8 - L
Substituting this equation into the area formula, we have:
Area = L × (8 - L)
Now, we need to find the maximum value of the area. This can be done by finding the vertex of the quadratic equation Area = L × (8 - L).
To find the vertex, we can either complete the square or use the fact that the vertex occurs at the midpoint of the x-values of the equation. In this case, because we are dealing with a quadratic equation with a maximum point, the midpoint of the x-values is the maximum point.
The midpoint value of L can be found by using:
Midpoint of L = (minimum value of L + maximum value of L) / 2
Since the minimum value of L is 0 (a rectangle with only width) and the maximum value of L would be half of the perimeter (16/2 = 8), the midpoint value of L is:
Midpoint of L = (0 + 8) / 2 = 4
Now we can substitute this midpoint value back into the equation to find the maximum area:
Area = 4 × (8 - 4)
Area = 4 × 4
Area = 16 square meters
Therefore, the largest area of a rectangular area that can be enclosed by a 16-meter fence is 16 square meters.