Gravel is being dumped from a conveyor belt at a rate of 35 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 8 ft high? (Round your answer to two decimal places.)

0.70 ft

Step-by-step explanation:

The volume V of a cone is given by

where r is the radius of the base and h is the height of the cone. Since the radius is half the diameter D, we can rewrite V as

But since the base diameter D is always equal to the cone's height, we can write the volume in terms of the height h as

Now gravel is being added to the conical pile at a rate of 35 cubic ft per minute so to find the rate at which the height of the pile is increasing, we are going to take the derivative of V with respect to time:

Rearranging the variables and solving for dh/dt, we get