SOLUTION: Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) whe

Question 1121706: Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains 1090 ft3 of grain?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the formula for the volume of a right circular cone is:

V = pi * r^2 * h / 3

V is the volume
pi is equal to 3.141592654
r is the radius
h is the height

you are given that the volume is 1090 cubic feet.

the formula becomes 1090 = pi * r^2 * h / 3

you are given that the diameter is always 3 times the height.

since the radius is always half the length of the diameter, then the radius is always 1.5 times the height.

replace r with 1.5 * h in the formula and the formula becomes:

1090 = pi * (1.5 * h)^2 * h / 3

this becomes 1090 = pi * 1.5^2 * h^2 * h / 3 which then becomes:

1090 = pi * 2.25 * h^3 / 3

divide both sides of this equation by (2.25 * pi) and multiply both sides of this equation by 3 to get:

(1090 * 3) / (2.25 * pi) = h^3

solve for h^3 to get h^3 = 462.6103679.

solve for h to get h = cube root of that = 7.734016969 feet.

since the radius is 1.5 times the height, then the radius is equal to 11.60102545 feet.

since the diameter is 2 times the length of the radius, then the diameter is equal to 23.20205091 feet.

to see if this is correct, replace the radius with 11.60102545 feet and replace the height with 7.734016969 feet and solve for the volume.

you will get V = pi * 11.60102545^2 * 11.60102545 / 3.

solve for V to get V = 1090.

your solution is that the pile 7.73 feet high when the volume is 1090 cubic feet.

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