Answer:
The maximum area is [tex]900\ ft^{2} [/tex]
Step-by-step explanation:
Let
x----> the length of rectangle
y---> the width of rectangle
we know that
The perimeter of rectangle is equal to
[tex]P=2(x+y)[/tex]
we have
[tex]P=120\ ft[/tex]
so
[tex]120=2(x+y)[/tex]
[tex]60=(x+y)[/tex]
[tex]y=60-x[/tex]------> equation A
Remember that
The area of rectangle is equal to
[tex]A=xy[/tex] -----> equation B
substitute equation A in equation B
[tex]A=x(60-x)[/tex]
[tex]A=-x^{2} +60x[/tex]
This is a vertical parabola open downward
The vertex is a maximum
The y-coordinate of the vertex of the graph is the maximum area of the garden and the x-coordinate is the length for the maximum area
using a graphing tool
The vertex is the point [tex](30,900)[/tex]
see the attached figure
Find the value of y
[tex]y=60-x[/tex] -----> [tex]y=60-30=30\ ft[/tex]
The dimensions of the rectangular garden is [tex]30\ ft[/tex] by [tex]30\ ft[/tex]
For a maximum area the garden is a square
The maximum area is [tex]900\ ft^{2} [/tex]
