The maximum area is achieved when the rectangle is a square of side 6 feet, with a domain, 0 < x < 12.
The perimeter available with Brayden is 24 feet.
The width of the rectangle is assumed to be x feet.
The length can be calculated using the formula:
2(length + width) = perimeter,
or, 2length + 2width = perimeter,
or, 2length = perimeter - 2width,
or, length = (1/2)(perimeter - 2 width).
Substituting the values, we get:
length = (1/2)(24 - 2x).
The area can be calculated using the formula:
Area = length*width.
Substituting the values, we get:
Area = (1/2)(24 - 2x)x = (1/2)x(24 - 2x).
Now, we need to maximize the area for the given perimeter.
For that, we differentiate the area function, with respect to its width x.
d(Area)/dx = (1/2)(24 - 2x) + (1/2)x(-2),
or, d(Area)/dx = 12 - x - x = 12 - 2x ... (i).
To check for the point of inflection, we equate this to zero, to get:
12 - 2x = 0,
or, 2x = 12,
or, x = 6.
To check whether this is maximum or minimum, we differentiate (i) with respect to x to get:
d²(Area)/dx² = -2 which is less than 0, implying area is maximum at x = 6.
Thus, the maximum area is achieved when the width is 6 feet.
Length = (1/2)(24 - 2x) = (1/2)(24 - 2*6) = (1/2)12 = 6.
Thus, the maximum area is achieved when the length is 6 feet.
The area function is, area = (1/2)x(24 - 2x).
We know that the area is always greater than 0, thus, we can show that:
(1/2)x(24-2x) > 0,
or, x(24 - 2x) > 0,
or, x(12 - x) > 0, which is true when 0 < x < 12.
Thus, the domain of the area function is 0 < x < 12.
Thus, the maximum area is achieved when the rectangle is a square of side 6 feet, with a domain, 0 < x < 12.
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