Sand falls from an overhead bin and accumulates in a conical pile with a radius

Question

Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always three times its height. Suppose the height of the pile increases at a rate of 2 cm/s when the pile is 14 cm high. At what rate is the sand leaving the bin at that instant? Let V and h be the volume and height of the cone, respectively. Write an equation that relates V and h and does not include the radius of the cone. (Type an exact answer, using as needed.) Differentiate both sides of the equation with respect to t. dV dt (Type an exact answer, using as needed.) dh dt The sand is leaving the bin at a rate of (Type an exact answer, using as needed.)

Explanation / Answer

from givev r = 3 * h

First get an equation for the volume of the cone in terms of its height

v = 1/3 * pi * r^2 * h

v = 1/3 * pi * (3 * h)^2 * h

v = 1/3 * pi * 9 * h^3

Take derivative of both sides

d/dt { v = 3 * pi * h^3 }

dv/dt = 9 * pi * h^2 * dh/dt

In your problem,

dh/dt =2cm/sd

h = 14 cm

Plug in and solve for dh/dt

dv/dt = 9* pi * 196 m^2 * 2

dv/dt = 3528 pi cm^3/s

sand is leaving at the rate of 3528 cm^3/s

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