1. **State the problem:** We need to find the surface area of a right triangular prism with legs 4 cm and 3 cm, and a prism length of 10 cm along the hypotenuse direction at an angle of 10°. 2. **Identify the shape and given data:** The base is a right triangle with legs $a=4$ cm and $b=3$ cm. The hypotenuse $c$ of the triangle is calculated by the Pythagorean theorem: $$c=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\text{ cm}$$ 3. **Calculate the area of the triangular base:** $$\text{Area}_{\triangle} = \frac{1}{2}ab = \frac{1}{2} \times 4 \times 3 = 6\text{ cm}^2$$ 4. **Determine the prism length:** The prism length is given as 10 cm along the hypotenuse direction at an angle of 10°. To find the actual length perpendicular to the base triangle, use the cosine of 10°: $$L = 10 \times \cos(10^\circ)$$ Calculate: $$L \approx 10 \times 0.9848 = 9.848\text{ cm}$$ 5. **Calculate the lateral surface area:** The perimeter of the triangular base is: $$P = 4 + 3 + 5 = 12\text{ cm}$$ The lateral surface area is perimeter times prism length: $$\text{Lateral Area} = P \times L = 12 \times 9.848 = 118.176\text{ cm}^2$$ 6. **Calculate total surface area:** The prism has two triangular bases and three rectangular lateral faces. $$\text{Total Surface Area} = 2 \times \text{Area}_{\triangle} + \text{Lateral Area} = 2 \times 6 + 118.176 = 12 + 118.176 = 130.176\text{ cm}^2$$ **Final answer:** $$\boxed{130.176\text{ cm}^2}$$