1. **State the problem:** We need to find the volume of a rectangular prism with dimensions length $10$ cm, width $4$ cm, and height $8$ cm, but it has two quarter-circle cutouts of radius $2$ cm on two adjacent edges. 2. **Calculate the volume of the full rectangular prism:** The volume $V_{prism}$ is given by the formula: $$V_{prism} = \text{length} \times \text{width} \times \text{height}$$ Substitute the values: $$V_{prism} = 10 \times 4 \times 8 = 320 \text{ cm}^3$$ 3. **Calculate the volume of one quarter-cylinder cutout:** Each cutout is a quarter of a cylinder with radius $r=2$ cm and height equal to the edge length it is cut from. - For the top-right corner (horizontal face), the quarter-cylinder height is the width $4$ cm. - For the bottom-right corner (vertical face), the quarter-cylinder height is the height $8$ cm. The volume of a full cylinder is: $$V_{cyl} = \pi r^2 h$$ Since the cutout is a quarter-cylinder, its volume is: $$V_{cutout} = \frac{1}{4} \pi r^2 h$$ 4. **Calculate volume of the first cutout (top-right corner):** $$V_{cutout1} = \frac{1}{4} \pi (2)^2 (4) = \frac{1}{4} \pi \times 4 \times 4 = 4\pi \text{ cm}^3$$ 5. **Calculate volume of the second cutout (bottom-right corner):** $$V_{cutout2} = \frac{1}{4} \pi (2)^2 (8) = \frac{1}{4} \pi \times 4 \times 8 = 8\pi \text{ cm}^3$$ 6. **Calculate total cutout volume:** $$V_{cutouts} = V_{cutout1} + V_{cutout2} = 4\pi + 8\pi = 12\pi \text{ cm}^3$$ 7. **Calculate the volume of the prism with cutouts:** $$V = V_{prism} - V_{cutouts} = 320 - 12\pi \text{ cm}^3$$ 8. **Final answer:** The volume of the prism with the two quarter-circle cutouts is: $$\boxed{320 - 12\pi \text{ cm}^3}$$