1. **State the problem:** Find the total surface area of a combined solid made by joining a triangular prism on top of a parallelogram prism along a congruent rectangular face. 2. **Identify given dimensions:** - Parallelogram prism base edges: 24 cm and 16 cm - Parallelogram prism height: 8 cm - Triangular prism rectangular face: 17.9 cm by 7 cm - Triangular prism height: 6 cm - The prisms are joined along the rectangular face 17.9 cm by 7 cm 3. **Calculate surface area of the parallelogram prism:** - Base area of parallelogram = base × height = $24 \times 8 = 192$ cm² - Perimeter of parallelogram base = $2(24 + 16) = 80$ cm - Lateral surface area = perimeter × height of prism = $80 \times 6 = 480$ cm² - Total surface area of parallelogram prism = $2 \times$ base area + lateral area = $2 \times 192 + 480 = 864$ cm² 4. **Calculate surface area of the triangular prism:** - Triangular base area: right triangle with legs 7 cm and 17.9 cm - Area = $\frac{1}{2} \times 7 \times 17.9 = 62.65$ cm² - Perimeter of triangular base = $7 + 17.9 + \sqrt{7^2 + 17.9^2}$ - Calculate hypotenuse: $\sqrt{49 + 320.41} = \sqrt{369.41} \approx 19.22$ cm - Perimeter = $7 + 17.9 + 19.22 = 44.12$ cm - Lateral surface area = perimeter × height = $44.12 \times 6 = 264.72$ cm² - Total surface area of triangular prism = $2 \times$ base area + lateral area = $2 \times 62.65 + 264.72 = 389.99$ cm² 5. **Calculate combined surface area:** - The prisms are joined along the rectangular face 17.9 cm by 7 cm, so this face is counted twice in individual surface areas. - Area of joined face = $17.9 \times 7 = 125.3$ cm² - Subtract one joined face area from total to avoid double counting: $$\text{Total surface area} = 864 + 389.99 - 125.3 = 1128.69 \text{ cm}^2$$ 6. **Final answer:** The total surface area of the combined solid is approximately **1128.69 cm²**.