Question

A tank in the shape of an inverted right circular cone has height 10 meters and radius 18 meters. It is filled with 5 meters of hot chocolate.
Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. Note: the density of hot chocolate is 1490kg/m^3

Answer 1

determine the centroid of inverted circular cone,
infinite small of mass,
dM = (rho)πr² dy (1)

use triangular proportion,

r/y = R/H
r = Ry/H (2)

substitute (2) into (1),

dM = (rho)π(Ry/H)² dy

static moment with respect to x -axis is

Sx = ∫ y dM

Y
Sx = ∫ y ρπ(Ry/H)² dy
0


Y
Sx = ¼ y⁴ ρπ(R/H)² |
0


Sx = ¼ Y⁴ ρπ(R/H)²

total mass is,

Y
M = ∫ ρπ(Ry/H)² dy
0


Y
M = ⅓ y³ ρπ(R/H)² |
0


M = ⅓ Y³ ρπ(R/H)²

height of inverted right circular cone centroid is

Y(o) = Sx/M = ¼ Y⁴ ρπ(R/H)²/{⅓ Y³ ρπ(R/H)²}

Y(o)= ¾ Y

W=mgh

W=(ρ_chocolate)(V_chocolate) g h

W=(ρ_chocolate)(⅓ πR² H) (g) Y(o)

W=(1540)(⅓ π * (4 * 17/5)² * 4)(9.81)(¾ * 4)

W = 8778441.47 J

Answer 2

W = F * d , integral [ 5, 10 ]
= (weight * upper volume ) * displacement
= ( 1490 * π r² * dy ) * ( 10 - y )
= [ 1490 * π ( 9y/5)² * dy ] * ( 10 - y )
= 1490 π ∫ ( 9y/5)² ( 10 -y) dy [ 5, 10]
= 8,689,056 J