1. **State the problem:** We need to find the volume of soil dug out to make a cylindrical well with depth (height) $21.5$ m and diameter $5$ m. 2. **Formula for volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height (depth) of the cylinder. 3. **Calculate the radius:** Diameter $d = 5$ m, so radius $r = \frac{d}{2} = \frac{5}{2} = 2.5$ m. 4. **Substitute values into the formula:** $$V = \pi \times (2.5)^2 \times 21.5$$ 5. **Calculate the squared radius:** $$(2.5)^2 = 6.25$$ 6. **Calculate the volume:** $$V = \pi \times 6.25 \times 21.5$$ 7. **Multiply the constants:** $$6.25 \times 21.5 = 134.375$$ 8. **Final volume:** $$V = \pi \times 134.375$$ 9. **Approximate using $\pi \approx 3.1416$:** $$V \approx 3.1416 \times 134.375 = 422.41$$ 10. **Round to the nearest tenth:** $$V \approx 422.4 \text{ cubic meters}$$ **Answer:** The volume of soil dug out is approximately **422.4 cubic meters**.