1. **State the problem:** We need to find the volume of a hexagonal prism with side length $4$ ft, height $3$ ft, and apothem $3.5$ ft. 2. **Formula for volume of a prism:** $$\text{Volume} = \text{Base Area} \times \text{Height}$$ 3. **Formula for area of a regular hexagon:** $$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$$ 4. **Calculate the perimeter of the hexagon:** Since the hexagon has 6 equal sides of length $4$ ft, $$\text{Perimeter} = 6 \times 4 = 24 \text{ ft}$$ 5. **Calculate the base area:** $$\text{Base Area} = \frac{1}{2} \times 24 \times 3.5 = 12 \times 3.5 = 42 \text{ ft}^2$$ 6. **Calculate the volume:** $$\text{Volume} = 42 \times 3 = 126 \text{ ft}^3$$ **Final answer:** The volume of the hexagonal prism is $126$ cubic feet.