1. **Problem statement:** Calculate the surface areas of the cuboid, the triangular prism, and then the total surface area of the toy house formed by attaching the prism on top of the cuboid. 2. **Surface area of the cuboid:** The cuboid has length $l=40$ cm, width $w=16$ cm, and height $h=12$ cm. Formula for surface area of a cuboid: $$SA = 2(lw + lh + wh)$$ Calculate each term: $$lw = 40 \times 16 = 640$$ $$lh = 40 \times 12 = 480$$ $$wh = 16 \times 12 = 192$$ Sum: $$640 + 480 + 192 = 1312$$ Surface area: $$SA = 2 \times 1312 = 2624 \text{ cm}^2$$ 3. **Surface area of the triangular prism:** The prism has triangular faces with base $b=20$ cm, height $h_t=12$ cm, and one side $s=16$ cm, and length $L=40$ cm. First, find the area of one triangular face: $$A_{triangle} = \frac{1}{2} b h_t = \frac{1}{2} \times 20 \times 12 = 120 \text{ cm}^2$$ The prism has two triangular faces, so total area of triangles: $$2 \times 120 = 240 \text{ cm}^2$$ Next, find the perimeter of the triangular face to calculate the lateral surface area. The triangle sides are $20$ cm (base), $16$ cm (given side), and the third side can be found using Pythagoras: Calculate the third side $c$: $$c = \sqrt{h_t^2 + (\frac{b}{2})^2} = \sqrt{12^2 + 10^2} = \sqrt{144 + 100} = \sqrt{244}$$ Approximate: $$\sqrt{244} \approx 15.62 \text{ cm}$$ Perimeter $P$: $$P = 20 + 16 + 15.62 = 51.62 \text{ cm}$$ Lateral surface area: $$LSA = P \times L = 51.62 \times 40 = 2064.8 \text{ cm}^2$$ Total surface area of prism: $$SA_{prism} = 240 + 2064.8 = 2304.8 \text{ cm}^2$$ 4. **Surface area of the toy house:** The toy house is the cuboid with the prism attached on top. The top face of the cuboid (area $40 \times 16 = 640$ cm$^2$) is covered by the prism, so it is not visible. Therefore, total surface area: $$SA_{house} = SA_{cuboid} + SA_{prism} - \text{area of cuboid top face}$$ Calculate: $$SA_{house} = 2624 + 2304.8 - 640 = 4288.8 \text{ cm}^2$$ **Final answers:** - Cuboid surface area: $2624$ cm$^2$ - Triangular prism surface area: $2304.8$ cm$^2$ - Toy house surface area: $4288.8$ cm$^2$