Let's denote the length of the rectangle as L and the width as W.
The perimeter of a rectangle is given by the formula P = 2L + 2W.
Given that the perimeter of the rectangle is 160 cm, we have:
2L + 2W = 160 --> L + W = 80 --> W = 80 - L
The area of a rectangle is given by the formula A = L * W.
So, the area of the rectangle is A = L * (80 - L).
Now, let's denote the side length of the square as S.
The perimeter of a square is given by the formula P = 4S.
Given that the perimeter of the square is 160 cm, we have:
4S = 160 --> S = 40
The area of a square is given by the formula A = S^2.
So, the area of the square is A = 40^2 = 1600 cm^2.
The difference between the areas of the rectangle and the square is given as 600 cm^2.
So, L * (80 - L) - 1600 = 600 --> L * (80 - L) = 2200
Expanding the equation, we get: 80L - L^2 = 2200
Rearranging the equation, we get: L^2 - 80L + 2200 = 0
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -80, and c = 2200.
Plugging in these values, we get:
L = (-(-80) ± √((-80)^2 - 4(1)(2200))) / (2(1))
L = (80 ± √(6400 - 8800)) / 2
L = (80 ± √(-2400)) / 2
Since the area of a rectangle cannot be negative, we discard the negative solution.
L = (80 + √(-2400)) / 2
L = (80 + √(2400 * (-1))) / 2
L = (80 + √(4800i^2)) / 2
L = (80 + √(4800i^2)) / 2
L = (80 + (i√4800)) / 2
L = (80 / 2) + (i√4800 / 2)
L = 40 + (i√4800 / 2)
L = 40 + 20i√3
Since the length of a rectangle cannot be imaginary, we discard this solution.
Therefore, there is no real solution for the length of the rectangle. Hence, the area of the rectangle cannot be determined.