Question

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 900 ft3 of grain?

Answer 1

the pile has a height H = 15.09 ft

Explanation:

The volume of a cone V is

V= 1/3*π*H*R²

where H= height , R=radius

if the diameter D=2*R is 3 times its height

D=3*H → 2*R=3*H → R=3/2*H

then

V= 1/3*π*H*R² = 1/3*π*H*(3/2*H)² = 1/3*π*H*9/4*H² = 1/12*π*H³

V= 1/12*π*H³

if the volume is V=900 ft³ , then solving for H

V= 1/12*π*H³

H = ∛(12*V/π) = ∛(12*900 ft³/π) = 15.09 ft

H = 15.09 ft

then the pile has a height H = 15.09 ft