Gravel is being dumped from a conveyor belt at a rate of 10 m^3/min The coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are in the ratio of 3 to 2 . How fast is the height of the pile increasing when the pile is 7 metres high?
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The volume of the pile is given by
V = (1/3)[(1/4)pi D^2 h], where D is the diameter of the cone and h is the height. It is given that dV/dt = 10, D:h = 3:2 (that is 2D = 3h)) and we are asked to find dh/dt when h =7.
By substituting D = (3/2)h, we have
V = (3/16)pi h^3]
Differentiate the equation with respect to t:
dV/dt = (9/16)pi h^2 (dh/dt)
By substituting the values for dV/dt and h, you can get the dh/dt