1. **State the problem:** We are given the volume $V = 2279.64$ cubic feet of a cone and the radius of the base $r = 11$ feet. We need to find the height $h$ of the cone. 2. **Recall the formula for the volume of a cone:** $$V = \frac{1}{3} \pi r^2 h$$ 3. **Substitute the known values:** $$2279.64 = \frac{1}{3} \pi (11)^2 h$$ 4. **Simplify inside the parentheses:** $$2279.64 = \frac{1}{3} \pi \times 121 \times h$$ 5. **Multiply constants:** $$2279.64 = \frac{121 \pi}{3} h$$ 6. **Solve for height $h$:** $$h = \frac{2279.64 \times 3}{121 \pi}$$ 7. **Calculate the numeric value:** Approximate $\pi \approx 3.1416$, $$h = \frac{6838.92}{379.94} \approx 18.01$$ 8. **Answer:** The height of the cone is approximately $\boxed{18.01}$ feet.