Final answer:
The height of a conical pile with a semi-vertical angle of 30° and base radius r is h = r. The volume of the cone is V = \(\frac{1}{3}\)\pi r^3, not V = \(\frac{\pi r^3}{\sqrt{33}}\) as mentioned in the question, which is a mistake.
Step-by-step explanation:
The question involves finding the height of a conical pile (part a) and the volume of the cone (part b) given the semi-vertical angle of 30°. For part a, since the semi-vertical angle is 30°, the height can be found by the relation in a 30-60-90 right triangle where the height (h) is the same as the radius (r), so h = r. For part b, the volume of a cone is V = \(\frac{1}{3}\)\pi r^2h. Since we know from part a that h = r, we can substitute h with r to get V = \(\frac{1}{3}\)\pi r^3. However, there is a typographical error in the question outlining the volume formula to be V = \(\frac{\pi r^3}{\sqrt{33}}\), which is incorrect and should be corrected as the former.