If ABCD is square, with side length x and a path of width y around the square, then the area of the path is:
4xy (the parts of the path bordering the square)
4y^2 (the square bits touching the corners of square ABCD).
The area of the path is 4xy + 4y^2
Square ABCD, with an area of 729, has a side length of 27, so x = 27.
The area of the path is given as 295, so:
4xy + 4y^2 = 295
4(27)y + 4y^2 = 295
108y + 4y^2 = 295
4y^2 + 108y - 295 = 0
Quadratic formula. . .
y = (-108 +- sqrt( 108^2 - 4(4)(-295) ) ) / 2(4)
y = (-108 +- sqrt(11664 + 4720) ) / 8
y = (-108 +- sqrt(16384) ) / 8
y = (-108 +- 128) / 8
It's a physical measurement, so it can't be negative. . .
y = (-108 + 128) / 8
y = 20/8
y = 2.5
The width of the path is y = 2.5, but what about the outer square (with the path surrounding the inner square)?
The outer square is x + 2y because the path sticks out on both sides of the inner square.
27 + 2(2.5) = 27 + 5 = 32
Answer: The path is 2.5 meters wide, the inner square is 27 meters wide, and the outer square is 32 meters wide.