Answer:
[tex]324 ft^2[/tex]
Step-by-step explanation:
The area of a triangle is given by the product between length and width:
[tex]A=Lw[/tex] (1)
where
L is the length
w is the width
The perimeter of the rectangle is given by
[tex]p=2L+2w[/tex]
In this problem, we know that the perimeter of the rectangle is fixed, and it is
[tex]p=72 ft[/tex]
So we have:
[tex]72=2L+2w[/tex]
Which can be rewritten as
[tex]w=36-L[/tex]
If we substitute this into the formula of the area, (1), we get:
[tex]A=L(36-L)=36L-L^2[/tex]
To maximize the area, we have to calculate its derivative and require it to be equal to zero:
[tex]\frac{dA}{dL}=0[/tex]
Calculating the derivative,
[tex]\frac{dA}{dL}=\frac{d}{dL}(36L-L^2)=36-2L[/tex]
And requiring it to be zero, we find:
[tex]36-2L=0\\L=\frac{36}{2}=18[/tex]
Which means also
[tex]w=36-L=36-18=18[/tex]
So,
L = 18 feet
w = 18 feet
So the maximum area is achieved when the rectangle has actually the shape of a square.
In such case, the area is:
[tex]A=18\cdot 18=324 ft^2[/tex]
So, this is the maximum area.
