1. **State the problem:** We need to find the surface area of a triangular prism with given dimensions: triangle sides 10 in., 11 in., and 11 in., and rectangular faces with dimensions 11 in. height and widths 6 in. and 8 in. 2. **Identify the shape and dimensions:** - The triangular base has sides 10 in., 11 in., and 11 in. - The prism height (length) is 8 in. - The rectangular faces correspond to the prism's height and the triangle's sides. 3. **Formula for surface area of a triangular prism:** $$\text{Surface Area} = 2 \times \text{Area of triangle base} + \text{Perimeter of triangle} \times \text{height of prism}$$ 4. **Calculate the area of the triangular base using Heron's formula:** - Semi-perimeter $s = \frac{10 + 11 + 11}{2} = 16$ in. - Area $= \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{16(16 - 10)(16 - 11)(16 - 11)}$ 5. **Calculate inside the square root:** $$16 \times 6 \times 5 \times 5 = 16 \times 150 = 2400$$ 6. **Area of triangle:** $$\sqrt{2400} = \sqrt{16 \times 150} = 4 \sqrt{150} \approx 4 \times 12.247 = 48.99 \text{ in}^2$$ 7. **Calculate the perimeter of the triangle:** $$10 + 11 + 11 = 32 \text{ in}$$ 8. **Calculate the lateral surface area:** $$32 \times 8 = 256 \text{ in}^2$$ 9. **Calculate total surface area:** $$2 \times 48.99 + 256 = 97.98 + 256 = 353.98 \text{ in}^2$$ 10. **Final answer:** The surface area of the triangular prism is approximately **354 square inches**.