Let the side length of each square base be x cm. Since the base of each prism is a square, the area of each base is x^2 square cm.
The total surface area of each prism consists of the areas of the two square bases and the four rectangular sides. Since the height of each prism is 90 cm, the area of each rectangular side is 90 * x square cm. Therefore, the total surface area of each prism is:
2(base area) + 4(side area) = 2(x^2) + 4(90x) = 2x^2 + 360x square cm
Since the surface area of each prism is K square cm, we have:
2x^2 + 360x = K
When the prisms are glued together along a square base, the resulting prism has a surface area of 92K square cm. The surface area of the glued prism is the sum of the surface areas of the two individual prisms minus the area of the glued base that is shared by both prisms. Since the base area is x^2 square cm and it is shared by both prisms, we subtract x^2 from the sum of the surface areas of the two individual prisms to get 92K:
2(K) - x^2 = 92K
2K - x^2 = 92K
-x^2 = 90K
x^2 = -90K
Since x represents a length, it must be positive. Therefore, the equation -x^2 = 90K has no valid solution, and there seems to be an error in the question or in the calculations.
