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# Grain is falling into from a conveyor into a conical pile at a rate of 8 cubic feet per minute.

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Grain is falling into from a conveyor into a conical pile at a rate of 8 cubic feet per minute. The diameter of the pile is four times the height of the pile.

A. Write an equation for the volume of the pile in terms of the height.

B. When the pile is 5 feet high, how has is the height of the pile changing?

C. When the pile is changing at a rate of 1.5 feet per minute, how tall is the pile?

Jan 13, 2020

#1
+11729
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Grain is falling into from a conveyor into a conical pile at a rate of 8 cubic feet per minute. The diameter of the pile is four times the height of the pile.

A. Write an equation for the volume of the pile in terms of the height.

B. When the pile is 5 feet high, how has is the height of the pile changing?

C. When the pile is changing at a rate of 1.5 cubic feet per minute, how tall is the pile????

Jan 13, 2020
edited by Omi67  Jan 13, 2020
#2
+109739
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B. When the  pile is 5 feet high, how fast is the height of the  pile changing

dv / dh  = 4pi h^2

So......

dV / dt   =  dV/ dh * dh /dt

8 ft^3   = 4pi h^2 * dh/dt

8 ft^3  = 4pi (5ft)^2  * dh/dt

8 ft^3 = 100pi ft^2 * dh /dt

8 / [ 100pi ]  = dh/dt

.08/pi  ft /min  = dh/dt

Jan 13, 2020
#3
+109739
+1

C. When the pile is changing at a rate of 1.5 feet per minute, how tall is the pile?

dh/ dt  =  4pi h^2

1.5 =  4 pi h^2

(1/5) / ( 4pi)  = h^2

√[ 1.5 / (4 pi) ]  = h ≈   .345 ft

Jan 13, 2020
edited by CPhill  Jan 13, 2020