Answer:
The maximum area is equal to [tex]20,000\ ft^{2}[/tex]
Step-by-step explanation:
Let
x ----> the length of the rectangular yard
y ----> the width of the rectangular yard
we know that
The perimeter is equal to
[tex]400=x+2y[/tex] --> remember that the fourth side will be against her deck
isolate the variable y
[tex]y=200-0.5x[/tex] -----> equation A
The area of the rectangular yard is equal to
[tex]A=xy[/tex] ----> equation B
substitute equation A in equation B
[tex]A=x(200-0.5x)\\ \\A=200x-0.5x^{2}[/tex]
The quadratic function is a vertical parabola open downward
The vertex is a maximum
The x-coordinate of the vertex represent the length of the rectangular yard for an maximum area
The y-coordinate of the vertex represent the maximum area of the rectangular yard
Using a graphing tool
The vertex is the point (200,20,000)
see the attached figure
therefore
The length of the rectangular yard is 200 ft
The width of the rectangular yard is [tex]y=200-0.5(200)=100\ ft[/tex]
The maximum area is equal to [tex]20,000\ ft^{2}[/tex]
