Answer:
The dimensions of the rectangle with the maximum area are 15 feet by 30 feet
Explanation:
Optimization
There is a rectangle of dimensions x and y as seen in the figure below.
The side of length y is divided into three equal parts to make three equal rectangles.
The area of the greater rectangle is:
A = xy
It has to be maximized given the condition that the total perimeter including the middle dividers is 120 feet.
The outer perimeter of the rectangle is 2x+2y. There are two additional lines with a length of x feet each. Thus, the total perimeter is:
P = 2x + 2y + 2x = 4x + 2y
It has to be equal to 120 feet:
4x + 2y = 120
Dividing by 2:
2x + y = 60
Solving for y:
y = 60 - 2x
Substituting into the area:
Taking the first derivative:
A'=60 - 4x
Equating to 0:
60 - 4x = 0
Solving for x:
x = 60/4 = 15
Thus x = 15 feet
Since y = 60 - 2x = 60 - 30 = 30
Thus y = 30 feet
The dimensions of the rectangle with the maximum area are 15 feet by 30 feet.