Answer:
The region is a square with a length side of 40.5 feet
Step-by-step explanation:
I will assume that the region is a rectangular area
we know that
The perimeter of the region is equal to
[tex]P=2(x+y)[/tex]
where
x is the length
y is the width
we have
[tex]P=162\ ft[/tex]
so
[tex]162=2(x+y)[/tex]
simplify
[tex]81=x+y[/tex]
[tex]y=81-x[/tex] ----> equation A
The area of the rectangular region is
[tex]A=xy[/tex] ----> substitute equation A in equation B
[tex]A=x(81-x)[/tex]
[tex]A=-x^2+81x[/tex]
This is the equation of a vertical parabola open downward
The vertex represent a maximum
The y-coordinate of the vertex represent the maximum area
The x-coordinate of the vertex represent the length for the maximum area
Convert the quadratic equation in vertex form
Factor -1
[tex]A=-(x^2-81x)[/tex]
Complete the square
[tex]A=-(x^2-81x+40.5^2)+40.5^2[/tex]
[tex]A=-(x^2-81x+40.5^2)+1,640.25[/tex]
Rewrite as perfect squares
[tex]A=-(x-40.5)^2+1,640.25[/tex]
The vertex is the point (40.5,1,640.25)
so
The maximum area is [tex]1.640.25\ ft^2[/tex]
The length for the maximum area is [tex]x=40.5\ ft[/tex]
Find the value of y (equation A)
[tex]y=81-40.5=40.5\ ft[/tex]
therefore
The region is a square with a length side of 40.5 feet
Answer:
20 by 40.5
Step-by-step explanation:
If the three equal vertical segments on the left, middle, and right of the region each have length L, and the segments on the top and bottom side of the region have length W, then we see that 3L+2W=162 (the total length of the fencing), so we find that
W=162−3L2
Therefore, in terms of the side length L, the area of the region is
L⋅(162−3L2)=162L−3L22
This is a quadratic function, where L is what we usually think of as x. To maximize the area we should find the x-value of its vertex. Recall that the x-value of the vertex of the parabola ax2+bx+c is given by the formula −b2a. So in this case, we have a=−32 and b=81, so the x-value of the vertex is
−b2a=−(81)2(−3/2)=27
So the optimal length of the left, middle, and right segments is 27, and the length of the top and bottom is
W=162−3(27)2=40.5
