Question

Sand pouring from a conveyor belt forms a conical pile the radius of which is 3/4 of the height. If the sand is filling up at a constant rate of 1/2m^3/min, at what rate is the height of the pile is growing 3 min after the pouring starts?

Answer 1

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Answer 2

since the radius is 3/4 the height,

The volume of the pile is thus

v = π/3 r^2 h = π/3 (3h/4)^2 h = 3π/16 h^3

dv/dt = 9π/16 h^2 dh/dt

now, plugging in your numbers,
at t=3, v = 3/2, so
3π/16 h^3 = 3/2
h = 2/∛π

so,

9π/16 4/∛π^2 dh/dt = 1/2
dh/dt = 2/9 ∛π m/min

Answer 3

thank you steve