1. **Stating the problem:** We have a prism with a cross-sectional shape of a sector of a circle. The sector has radius $r$ cm, central angle $50^\circ$, the prism length is 20 cm, and the height (thickness) is 8 cm. We want to find the volume of this prism. 2. **Formula used:** The volume $V$ of a prism is given by: $$V = \text{Area of cross-section} \times \text{length}$$ 3. **Area of the sector cross-section:** The area $A$ of a sector with radius $r$ and central angle $\theta$ (in degrees) is: $$A = \frac{\theta}{360} \times \pi r^2$$ 4. **Calculate the area of the sector:** Given $\theta = 50^\circ$, the area is: $$A = \frac{50}{360} \times \pi r^2 = \frac{5}{36} \pi r^2$$ 5. **Calculate the volume of the prism:** The prism length is 20 cm, so: $$V = A \times 20 = \frac{5}{36} \pi r^2 \times 20 = \frac{100}{36} \pi r^2 = \frac{25}{9} \pi r^2$$ 6. **Incorporate the height (thickness) of 8 cm:** Since the prism has thickness 8 cm, the volume is: $$V = \text{Area of sector} \times \text{length} \times \text{height} = \frac{5}{36} \pi r^2 \times 20 \times 8$$ Simplify: $$V = \frac{5}{36} \times 20 \times 8 \times \pi r^2 = \frac{5 \times 20 \times 8}{36} \pi r^2 = \frac{800}{36} \pi r^2 = \frac{200}{9} \pi r^2$$ **Final answer:** $$\boxed{V = \frac{200}{9} \pi r^2 \text{ cubic centimeters}}$$ This formula gives the volume of the prism in terms of the radius $r$ of the sector cross-section.