SOLUTION: Gravel is being dumped from a conveyor belt at a rate of 15 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are alway

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Question 1127031: Gravel is being dumped from a conveyor belt at a rate of 15 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 5 ft high? (Round your answer to two decimal places.)
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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The volume of a cone   V = %281%2F3%29%2Api%2Ar%5E2%2Ah.


Since we are given that  h = 2r (where 2r is the diameter), 


    V = %281%2F3%29%2Api%2A%28h%2F2%29%5E2%2Ah = %281%2F3%29%2Api%2A%281%2F4%29%2Ah%5E3 = %28pi%2F12%29%2Ah%5E3.    (1)


The rate of the volume change in time is given;  it is 15 %28ft%29%5E3%2F%28min%29;  therefore


15 = %28dV%29%2F%28dt%29 = %28%28dV%29%2F%28dh%29%29%2A%28%28dh%29%2F%28dt%29%29 = (calculate %28dV%29%2F%28dh%29 from the formula (1)) = %28pi%2F4%29%2Ah%5E2.%28dh%29%2F%28dt%29 = (substitute here h= 5 ft) = %28pi%2F4%29%2A25.%28dh%29%2F%28dt%29.


It implies  %28dh%29%2F%28dt%29 = 15%2F%28%28pi%2F4%29%2A25%29 = %283%2A4%29%2F%285%2Api%29 = 12%2F%285%2Api%29 = 0.76 ft/min.  (rounded)


Answer.  At this moment, the height of the cone is increasing at the rate of  0.76 ft/min. (approximately)

Solved.