Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always four times its height. Suppose the height of the pile increases at a rate of 1 cm/s when the pile is 13 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 n as needed.)Differentiate both sides of the equation with respect to t.dVdh%3Ddtdt(Type an exact answer, using n as needed.)The sand is leaving the bin at a rate of(Type an exact answer, using a as needed.)

Let r be the radius and h be the height of the cone. Given that the radius is 3 times the height of the cone, that is, r=3h.The formula for the volume of a cone is: V=13πr2h Put r=3h in the above formula. V=13π3h2h=3πh3Since the height of the cone can be observed as a function of t, where represents time, denote the volume as follows: V=V(h(t)) Differentiate both sides of the equation with respect to t. dVdt=dVdh·dhdt Since the height of the pile increases at a rate of 1 cm/s, when the pile is 13 cm high gives: dhdt=1 cm/s Also, dVdh=ddh3πh3=9πh2. Therefore, dVdt=9πh2·dhdt Plug the value of dhdt. dVdt=9πh2·1 cm/s=9πh2, where h=13 cm. Substitute h=12 cm in the previous equation. dVdt=9π×132=1521π cm3/s Thus the sand is leaving the bin at a rate of 1521π cm3/s.

Answered: Sand falls from an overhead bin and…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always four times its height. Suppose the height of the pile increases at a rate of 1 cm/s when the pile is 13 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 n as needed.)
Differentiate both sides of the equation with respect to t.
dV
dh
%3D
dt
dt
(Type an exact answer, using n as needed.)
The sand is leaving the bin at a rate of
(Type an exact answer, using a as needed.)

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