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 9 ft high? (Round your answer to two decimal places.)

Question

Mathematics

Orion Holland

13 June, 11:58

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 9 ft high? (Round your answer to two decimal places.)

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Answers (1)

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Jace Pham · Answer 1

dh/dt ≈ 0.55 ft/min

Step-by-step explanation:

The volume is given by the formula ...

V = (1/3) πr²h

We have r = h/2, so the volume as a function of height is ...

V = (1/3) π (h/2) ²h = (π/12) h³

Then the rates of change are related by ...

dV/dt = (π/4) h²·dh/dt

dh/dt = (4·dV/dt) / (πh²) = 4 (35 ft³/min) / (π (9 ft) ²)

dh/dt ≈ 0.55 ft/min