Given that grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height.
So if D is the diameter and h is the height of the conical pile then we can write:
D=3h
We know that diameter = 2r, where r is the radius
2r=3h
[tex]r=\frac{3h}{2}[/tex]
Volume of conical pile is given by formula
[tex] V=\frac{1}{3}\pi r^2h[/tex]
Given that volume is 1110 cubic feet.
Now plug the values of Volume and r into equation of volume
[tex] 1110=\frac{1}{3}\pi (\frac{3h}{2})^2h[/tex]
[tex] 1110=\frac{1}{3}\pi\cdot\frac{9h^2}{4}\cdot h[/tex]
[tex] 1110=\pi\cdot\frac{3h^3}{4}[/tex]
[tex] 1110\cdot\frac{4}{3\pi}=h^3[/tex]
[tex] 471.098631552=h^3[/tex]
take cube root of both sides
7.78103342467=h
Hence height is approx 7.78 feet.
