Answer:
[tex]P=40\cdot 20=800[/tex]
Step-by-step explanation:
We know that Manuel decides to build a fence to enclose an area for his dogs to play. The area is to be rectangular in shape. We know that only 3 sides need to be fenced, because his house is to be one border of the rectangle.
We conclude that the the maximum area is:
[tex]P=40\cdot 20=800[/tex]
So the two shorter sides will be 20 feet each, the longer side will be 40 feet.
The dimension that will give the dogs the maximum area area 20 feet and 40 feet.
He builds a fence enclose an area. The area to be enclosed is rectangular in shape.
He has 80 feet of fencing . He only wants to fence 3 sides because his house completes the border of the rectangle. Therefore,
let
x = side parallel to the border of the house
y = sides perpendicular to the border of the house
Therefore,
x + 2y = 80(perimeter)
x = 80 - 2y
area = xy
area = y(80 - 2y) = 80y - 2y²
area = -2y² + 80y
The graph is a parabola and it opens downward because the leading coefficient is less than zero. The leading coefficient is -2.
The maximum occurs at the x coordinates of the vertex. Therefore,
x = h = [tex]-\frac{b}{2a}[/tex]
where
b = 80
a = -2
x = [tex]-\frac{80}{-2 (2)}[/tex]
x = [tex]\frac{80}{4}[/tex]
x = 20
To maximize the area, the 2 sides perpendicular to the border of the house should each have a length of x = 20 ft and the side parallel to the border of the house should have length y = 40 ft.
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