1. **State the problem:** We have a rectangular prism with edges summing to 28 cm and total surface area 13 cm². We want to find the length of the diagonal $EC$. 2. **Define variables:** Let the edges be $x$, $y$, and $z$. 3. **Sum of edges:** A rectangular prism has 12 edges: 4 edges of length $x$, 4 of $y$, and 4 of $z$. So, $$4(x + y + z) = 28 \implies x + y + z = 7$$ 4. **Surface area:** The surface area is the sum of areas of 6 faces: $$2(xy + yz + zx) = 13 \implies xy + yz + zx = \frac{13}{2} = 6.5$$ 5. **Diagonal length:** The diagonal $EC$ connects opposite vertices, so by the 3D Pythagorean theorem, $$EC = \sqrt{x^2 + y^2 + z^2}$$ 6. **Find $x^2 + y^2 + z^2$:** Use the identity $$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) $$ Substitute known values: $$7^2 = x^2 + y^2 + z^2 + 2(6.5)$$ $$49 = x^2 + y^2 + z^2 + 13$$ $$x^2 + y^2 + z^2 = 49 - 13 = 36$$ 7. **Calculate diagonal:** $$EC = \sqrt{36} = 6$$ **Final answer:** The length of diagonal $EC$ is $6$ cm.