1. **State the problem:** Calculate the volume of a pentagonal prism with given edge lengths. 2. **Formula:** The volume $V$ of a prism is given by: $$V = B \times h$$ where $B$ is the area of the base and $h$ is the height of the prism. 3. **Identify given values:** - Height $h = 3$ cm - Base is a pentagon with sides including 4.5 cm, 2 cm, 3.5 cm, 3.5 cm, and a perpendicular segment of 2 cm forming a right angle. 4. **Calculate the area of the pentagonal base:** The pentagon can be divided into simpler shapes. Given the right angle and segments, we can split the base into a rectangle and a triangle. - Rectangle area: $4.5 \text{ cm} \times 2 \text{ cm} = 9$ cm$^2$ - Triangle area: The triangle formed by sides 3.5 cm, 3.5 cm, and the perpendicular 2 cm can be considered as a right triangle with legs 3.5 cm and 2 cm. Triangle area: $$\frac{1}{2} \times 3.5 \times 2 = 3.5 \text{ cm}^2$$ 5. **Total base area:** $$B = 9 + 3.5 = 12.5 \text{ cm}^2$$ 6. **Calculate volume:** $$V = B \times h = 12.5 \times 3 = 37.5 \text{ cm}^3$$ **Final answer:** The volume of the pentagonal prism is $37.5$ cubic centimeters.