The piece of string, 12 m long, has to be used to form the border around a rectangle such that the length is 3/2 times the width. To maximize the area of the rectangle, it is only required that the whole length of string be used to create the border. Let the width of the rectangle be x, the length of the rectangle is 1.5*x
2*(x + 1.5*x) = 2*2.5*x = 5x = 12
=> x = 12/5
The with of the rectangle is 12/5 m and the length of the rectangle is 18/5.
The area of a rectangle is 216/25 m^2.
The area of the largest rectangle that can be created is 8.64 m^2.