Answer:
The side width that produce the maximum area will be 50 meters
Step-by-step explanation:
The correct question is
Marquise has 200 meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width w (in meters) is modeled by: A ( w ) = − w 2 + 1 0 0 w. What side width will produce the maximum garden area?
we have
[tex]A(w)=-w^2+100w[/tex]
where
A(w) ---> represent the garden's area in square meters
w ----> represent the garden width in meters
we know that
The given quadratic equation represent a vertical parabola open downward (the leading coefficient is negative)
The vertex represent a maximum
so
The x-coordinate of the vertex represent the width of the garden for the maximum area
The y-coordinate of the vertex represent the maximum area of the garden
Convert the quadratic equation into vertex form
[tex]A(w)=-w^2+100w[/tex]
Factor the leading coefficient -1
[tex]A(w)=-(w^2-100w)[/tex]
Complete the square
[tex]A(w)=-(w^2-100w+50^2)+50^2[/tex]
[tex]A(w)=-(w^2-100w+2,500)+2,500[/tex]
Rewrite as perfect squares
[tex]A(w)=-(w-50)^2+2,500[/tex] ----> equation in vertex form
The vertex is the point (50,2,500)
Therefore
The maximum area of the garden is 2,500 square meters
The side width that produce the maximum area will be 50 meters (x-coordinate of the vertex)
