1. **State the problem:** Find the surface area of a composite solid made of two rectangular prisms joined together. 2. **Identify dimensions:** - Prism 1: $13 \text{ cm} \times 7 \text{ cm} \times 5 \text{ cm}$ - Prism 2: $5 \text{ cm} \times 5 \text{ cm} \times 2 \text{ cm}$ 3. **Formula for surface area of a rectangular prism:** $$SA = 2(lw + lh + wh)$$ where $l$, $w$, and $h$ are the length, width, and height. 4. **Calculate surface area of Prism 1:** $$SA_1 = 2(13 \times 7 + 13 \times 5 + 7 \times 5)$$ $$= 2(91 + 65 + 35) = 2(191) = 382 \text{ cm}^2$$ 5. **Calculate surface area of Prism 2:** $$SA_2 = 2(5 \times 5 + 5 \times 2 + 5 \times 2)$$ $$= 2(25 + 10 + 10) = 2(45) = 90 \text{ cm}^2$$ 6. **Calculate the area of the overlapping face:** The two prisms are joined along a face of size $5 \text{ cm} \times 5 \text{ cm}$. $$A_{overlap} = 5 \times 5 = 25 \text{ cm}^2$$ 7. **Calculate total surface area of the composite solid:** $$SA = SA_1 + SA_2 - 2 \times A_{overlap}$$ We subtract twice the overlapping area because it is counted in both prisms' surface areas but is not exposed. $$SA = 382 + 90 - 2 \times 25 = 472 - 50 = 422 \text{ cm}^2$$ **Final answer:** $$\boxed{422 \text{ cm}^2}$$