1. **State the problem:** We have a rectangular prism ABCDEFGH with the sum of all edge lengths equal to 28 cm and total surface area equal to 13 cm². We need to find the length of the diagonal EC. 2. **Define variables:** Let the prism have side lengths $x$, $y$, and $z$. 3. **Sum of edges:** A rectangular prism has 12 edges: 4 edges of length $x$, 4 of length $y$, and 4 of length $z$. The sum of all edges is: $$4(x + y + z) = 28$$ Divide both sides by 4: $$\cancel{4}(x + y + z) = \cancel{4}7$$ So, $$x + y + z = 7$$ 4. **Surface area:** The total surface area is the sum of areas of all 6 faces: $$2(xy + yz + zx) = 13$$ Divide both sides by 2: $$\cancel{2}(xy + yz + zx) = \cancel{2}6.5$$ So, $$xy + yz + zx = 6.5$$ 5. **Find the diagonal EC:** The diagonal EC connects opposite corners of the prism. Its length is given by the 3D distance formula: $$EC = \sqrt{x^2 + y^2 + z^2}$$ 6. **Express $x^2 + y^2 + z^2$ in terms of $x+y+z$ and $xy+yz+zx$:** Recall the identity: $$(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)$$ Rearranged: $$x^2 + y^2 + z^2 = (x + y + z)^2 - 2(xy + yz + zx)$$ 7. **Substitute known values:** $$x^2 + y^2 + z^2 = 7^2 - 2(6.5) = 49 - 13 = 36$$ 8. **Calculate diagonal length:** $$EC = \sqrt{36} = 6$$ **Final answer:** The length of diagonal EC is **6 cm**.