1. **State the problem:** We need to find the volume of a cylinder with height $h=12$ feet and radius $r=4$ feet. 2. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ This formula calculates the volume by multiplying the area of the circular base ($\pi r^2$) by the height $h$. 3. **Substitute the given values:** $$V = \pi \times 4^2 \times 12$$ 4. **Calculate the square of the radius:** $$4^2 = 16$$ So, $$V = \pi \times 16 \times 12$$ 5. **Multiply the constants:** $$16 \times 12 = 192$$ So, $$V = \pi \times 192$$ 6. **Calculate the volume using $\pi \approx 3.1416$:** $$V \approx 3.1416 \times 192 = 602.88$$ 7. **Round to the nearest hundredth:** The volume is approximately $602.88$ cubic feet. **Final answer:** $$\boxed{602.88 \text{ cubic feet}}$$