The length and width of the rectangular pen that optimizes the area are 400 feet and 200 feet, respectively, according to the perimeter of a rectangle.
The perimeter of a shape is the sum of the lengths of all its edges. For example, a triangle has three edges, hence its perimeter is equal to the sum of those three lengths.The perimeter of a form can be calculated by adding the lengths of all its sides. A perimeter example is what? A field with a length of 24 yards and a width of 15 yards, for instance, will have a perimeter of 78 yards.
calculation
In order to construct a hog pen, I have 800 feet of fencing.
Assume that "l" represents the rectangle's length and "w" its width.
One of your properties borders a river that I own.
I can secure my property with fencing on its three sides.
because of this, the fencing's necessary perimeter p = l + 2w
The fencing's perimeter is now equal to
⇒ l + 2w = 800
= l = 800 - 2w
now the rectangle's area = A = lw
Therefore, A = l * w = w ( 800 - 2w ) = 800w - [tex]2w^{2}[/tex]
By diffusing the area equation, we can figure out the rectangle's greatest area.
Therefore, A' = 800 - 4w
Again, A" = -4
The rectangle has the largest area since the second order differentiation is constant negative.
Put simply, we obtain: A' = 0 , A' = 800- 4w
⇒800- 4w = 0
⇒ 4 w = 800
⇒ w = 200
So the rectangle has a 200-foot width.
The rectangle's length is currently:
l = 800 - 2w = 800 - 2 ( 200 ) = l = 400 feet
The length and width of the rectangular pen that optimizes the area are 400 feet and 200 feet, respectively, according to the perimeter of a rectangle.
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