1. The problem involves a pentagonal prism with a base edge length of 4.5 cm and a prism length (height) of 12 cm. 2. To find the volume of the prism, we need the area of the pentagonal base and multiply it by the prism's length. 3. The area $A$ of a regular pentagon with side length $s$ is given by: $$A = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} s^2$$ 4. Substitute $s = 4.5$ cm: $$A = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} (4.5)^2$$ 5. Calculate inside the square root: $$5 + 2\sqrt{5} \approx 5 + 2 \times 2.236 = 5 + 4.472 = 9.472$$ 6. Then: $$5 \times 9.472 = 47.36$$ 7. So: $$A = \frac{1}{4} \sqrt{47.36} \times 20.25$$ 8. Calculate the square root: $$\sqrt{47.36} \approx 6.88$$ 9. Then: $$A = \frac{1}{4} \times 6.88 \times 20.25 = 1.72 \times 20.25 = 34.83 \text{ cm}^2$$ 10. The volume $V$ of the prism is: $$V = A \times \text{length} = 34.83 \times 12 = 417.96 \text{ cm}^3$$ Final answer: The volume of the pentagonal prism is approximately $417.96$ cubic centimeters.