Answer:
The dimensions of the wall are 100 ft x 100 ft
Step-by-step explanation:
we know that
The perimeter of a rectangular wall is
[tex]P=2(x+y)[/tex]
where
x is the length
y is the width
we have
[tex]P=400\ ft[/tex]
so
[tex]400=2(x+y)[/tex]
simplify
[tex]200=x+y[/tex]
[tex]y=200-x[/tex] ----> equation A
The area of a rectangular wall is equal to
[tex]A=xy[/tex] ----> equation B
substitute equation A in equation B
[tex]A=x(200-x)\\A=-x^2+200x[/tex]
This is the equation of a vertical parabola open downward (because the leading coefficient is negative)
The vertex represent a maximum
Convert the quadratic equation in vertex form
[tex]A=-x^2+200x[/tex]
Factor -1
[tex]A=-(x^2-200x)[/tex]
Complete the square
[tex]A=-(x^2-200x+100^2)+100^2[/tex]
[tex]A=-(x^2-200x+10,000)+10,000[/tex]
Rewrite as perfect squares
[tex]A=-(x-100)^2+10,000[/tex] ----> equation in vertex form
The vertex is the point (100,10,000)
The x-coordinate of the vertex represent the length of the wall for a maximum area
so
[tex]x=100\ ft[/tex]
Find the value of y
equation A
[tex]y=200-(100)=100\ ft[/tex]
therefore
The dimensions of the wall are 100 ft x 100 ft
