The length that results in the maximum area is 39 feet.
Explanation
Lets assume, length of the rectangle is [tex] x [/tex] feet and width of the rectangle is [tex] y [/tex] feet.
As the length and width must add up to 78 feet, so the equation will be...
[tex] x+y=78 ................................................(1) [/tex]
Solving equation (1) for y : [tex] y= 78-x [/tex]
Now the area of the rectangle,
[tex] A = x*y\\\\ A= x(78-x)\\\\ A= 78x-x^2\\\\ A= -x^2 +78x...................(2) [/tex]
[tex] A [/tex] will be maximum when [tex] \frac{dA}{dx} = 0 [/tex]
Now taking derivative of equation(2) with respect to [tex] x [/tex]......
[tex] \frac{dA}{dx}= -2x+78 [/tex]
If [tex] \frac{dA}{dx} =0 [/tex], then
[tex] -2x+78=0\\\\ -2x=-78\\\\ x=\frac{-78}{-2}=39 [/tex]
If [tex] x= 39 [/tex], then [tex] y= 78-x = 78-39=39 [/tex]
So, both length and width will be 39 feet for getting maximum area.
The length that results in the maximum area is 39 feet.
