1. **State the problem:** We need to find the surface area of a triangular prism with a triangular base having sides 7 m, 4 m, and 3 m, and the prism length (height) is 11 m. 2. **Formula for surface area of a triangular prism:** The surface area $SA$ is given by: $$SA = 2 \times \text{Area of triangular base} + \text{Perimeter of base} \times \text{Length of prism}$$ 3. **Calculate the area of the triangular base:** Since the triangle has sides 3 m, 4 m, and 7 m, check if it is a right triangle. Using Pythagoras theorem: $3^2 + 4^2 = 9 + 16 = 25$ and $7^2 = 49$, so it is not a right triangle. Use Heron's formula: $$s = \frac{3 + 4 + 7}{2} = 7$$ $$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{7(7-3)(7-4)(7-7)} = \sqrt{7 \times 4 \times 3 \times 0} = 0$$ This means the sides 3, 4, and 7 do not form a valid triangle (area zero). However, the problem states the 3 m side is the height from the right angle to the base 4 m, so the triangle is right angled with legs 3 m and 4 m. Area of right triangle: $$\text{Area} = \frac{1}{2} \times 3 \times 4 = 6 \text{ m}^2$$ 4. **Calculate the perimeter of the base:** $$P = 3 + 4 + 7 = 14 \text{ m}$$ 5. **Calculate the lateral surface area:** $$\text{Lateral area} = P \times \text{length} = 14 \times 11 = 154 \text{ m}^2$$ 6. **Calculate total surface area:** $$SA = 2 \times 6 + 154 = 12 + 154 = 166 \text{ m}^2$$ **Final answer:** The surface area of the triangular prism is $166$ square meters.