1. **Stating the problem:** We have a right-angled trapezoidal prism with bases 4 cm and 3 cm, height 12 cm, and volume 456 cm³. We need to calculate the surface area correct to 1 decimal place and find the unknown length $x$ given the volume. 2. **Calculate the area of the trapezium base:** The area $A$ of a trapezium is given by $$A = \frac{1}{2} (a + b) h$$ where $a=4$ cm, $b=3$ cm, and height $h=12$ cm. $$A = \frac{1}{2} (4 + 3) \times 12 = \frac{1}{2} \times 7 \times 12 = 42 \text{ cm}^2$$ 3. **Calculate the surface area:** The prism has two trapezium bases and three rectangular faces. The rectangular faces correspond to the sides of the trapezium and the length $x$ (depth) of the prism. - The two trapezium bases contribute $2 \times 42 = 84$ cm². - The three rectangular faces have areas: - $4 \times x$ - $3 \times x$ - $12 \times x$ Total surface area $S$ is: $$S = 84 + (4x + 3x + 12x) = 84 + 19x$$ 4. **Calculate $x$ using the volume:** Volume $V$ of the prism is the area of the base times the length $x$: $$V = A \times x = 42 \times x = 456$$ Solve for $x$: $$x = \frac{456}{42} = 10.8571... \approx 10.9 \text{ cm}$$ 5. **Calculate the surface area with $x=10.9$ cm:** $$S = 84 + 19 \times 10.9 = 84 + 207.1 = 291.1 \text{ cm}^2$$ **Final answers:** - Length $x \approx 10.9$ cm - Surface area $\approx 291.1$ cm² (correct to 1 decimal place)