1. **Problem statement:** We have two identical rectangular prisms, each with height $90$ cm and a square base with side length $s$ cm. Each prism has surface area $K$ cm$^2$. When glued together along one square base, the new prism's surface area is $\frac{92}{47}K$ cm$^2$. We need to find $s$. 2. **Formula for surface area of one prism:** Each prism has 5 faces: 2 square bases and 4 rectangular sides. Surface area $K = 2s^2 + 4(s \times 90) = 2s^2 + 360s$. 3. **Surface area of glued prism:** When glued along one square base, that base is no longer exposed, so the combined prism has: - 1 square base (bottom) - 1 square base (top) - 4 rectangular sides (height $180$ cm because prisms are stacked) Surface area of glued prism = $2s^2 + 4(s \times 180) = 2s^2 + 720s$. 4. **Given relation:** $$2s^2 + 720s = \frac{92}{47} (2s^2 + 360s)$$ 5. **Solve the equation:** Multiply both sides: $$2s^2 + 720s = \frac{92}{47} \times 2s^2 + \frac{92}{47} \times 360s$$ Rewrite: $$2s^2 + 720s = \frac{184}{47}s^2 + \frac{33120}{47}s$$ Multiply both sides by 47 to clear denominator: $$47(2s^2 + 720s) = 184s^2 + 33120s$$ $$94s^2 + 33840s = 184s^2 + 33120s$$ Bring all terms to one side: $$0 = 184s^2 + 33120s - 94s^2 - 33840s$$ $$0 = 90s^2 - 720s$$ 6. **Factor:** $$0 = 90s(s - 8)$$ 7. **Solutions:** $s = 0$ (not valid for side length), or $s = 8$ cm. **Final answer:** The side length of each square base is $\boxed{8}$ cm.