calculus work problem - Liquid pumped out of a right circular cone?

Question

A tank in the shape of an inverted right circular cone has height 10m and radius 8m. It is filled to a depth of 3m with 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 1460 kg/m^3

I'm in dire straits! I need a solution immediately!

1st answer is not correct! Integration should be over the interval [7, 10]. 2nd answer: I can't understand what you're saying!

Answer 1

P.E. = mgh = ρVgh
V = (1/3)πr^2h = (1/3)π(3*8/10)^2(3)
dV = πr^2dh
dm = ρπr^2dh
dE = ρπr^2hdh
r = 8h/10 (by similar triangles)
dE = (64/100)ρπgh^3dh
PE = (64/400)ρπgh^4
(with respect to the bottom of the cone)
P.E.1 = (64/400)(1460)π(9.801)(3)^4
Neglecting the diameter of the outlet, and where the chocolate goes,
P.E.2 = ρVgh = (1460) (1/3) π (3*8/10)^2 (3) (9.801) (10)
P.E.2 = (1460) (1/3) π (3*8/10)^2 (3) (9.801) (10)
∆PE = (1460) (1/3) π(3*8/10)^2 (3) (9.801) (10) - (64/400) (1460) π (9.801) (3)^4
∆PE = (64/100) (1460) π (9) (9.801) (10) - (64/100) (1460) (1/4) π (9.801) (81)
W = ∆PE = (64/100) (1460) (9.801) (9) π [(10) - (1/4)(9)]
W = 20,4751 J, or 20.4751 kJ

Answer 2

Leave the current height of chocolate as x...

then...

the radius of the top of choco-cone is r=8*(x/10)=4*x/5

now, let's work out the delta_V (partial volume with delta_x; here delta_x is partial height of chocolate which represents the pumping chocolate per unit-time...)

delta_V=pi*(r^2)*delta_x (m^3)

delta_m=delta_v*q=delta_V*1460kg/m^3
=pi*(r^2)*delta_x*1460 (kg)
=pi*(16*x*x/25)*delta_x*1460

delta_A=F*s=delta_m*g*(10-x)
(here delta_A is the partial work for pumping chocolate per unit-time)

so,

A=Calculus(3 to 0) (pi*(16*x*x/25)*1460*g*(10-x)) dx
(here delta_x is converted into dx)

=(pi*934.4*g)*Calculus(3 to 0) [x^2*(10-x)] dx
=(pi*934.4*g)*110.25=
=103017.6*pi*g (J)

answer A=103017.6*pi*g(J) (here g=9.8)

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Answer 3

the simplest way

let's consider the weight center of fulfilled cone then volume of its and then calculate the work energy =volume*density*delta h
delta h=(10-3)+1/3(3)=8m
volume=your job

Answer 4

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