Final answer:
The question asks for the altitude of a conical pile of sand given the rate of sand falling and the relationship between the cone's diameter and altitude. To solve it, we would use the cone's volume formula, differentiate with respect to time, and solve for the altitude in terms of the rate of volume increase. However, the question lacks sufficient data to provide a numeric answer.
Step-by-step explanation:
The question is concerned with the rate at which sand is forming a conical pile, and it provides information about the relationship between the diameter of the cone's base and its altitude. To find the altitude of the cone, we need to use the formula for the volume of a cone, which is V = (1/3)πr^2h, where V is the volume, r is the radius of the base, and h is the height or altitude of the cone. Additionally, we are given that the diameter is three times the altitude (d = 3h), thus the radius is half the diameter (r = 1.5h).
Substituting the radius in terms of the altitude into the volume formula gives us a new equation solely in terms of the altitude. By differentiating this volume with respect to time, we could get an equation that includes the rate at which the volume of the sand pile is increasing, which is given as 8 cubic feet per minute. We would then solve this equation for the altitude h to get the value that we need.
However, the question as stated does not provide sufficient data to actually calculate the altitude, as the current volume or the time at which we are observing the pile is not given. Thus, we cannot provide a numeric answer to this particular question without additional information.