Question

A tank in the shape of an inverted right circular cone has height 4 meters and radius 2 meters. It is filled with 3 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is δ=1020 kg/m3.δ=1020 kg/m3. Your answer must include the correct units.

Answer 1

The required work to empty the the tank is 247401 J.

Step-by-step explanation:

Given that,

A tank is in the shape of right circular cone.

The radius of the tank= 2 m

The height of the tank = 4 m

The relation between the radius and the height is

`(r)/(h)=(2)/(4)`

`\Rightarrow r= \frac12 h`

Let h be the height of hot of chocolate any time t.

The volume of a cone is
`= \frac 13 \pi r^2 h`

`=\frac13 \pi (\frac h2)^2h`

`=\frac16 \pi h^3`

The volume of the chocolate is
`=\frac16 \pi h^3`

The mass of the chocolate is(M)= Density × Volume

`=1020(\frac16 \pi h^3)` Kg

`=170\pi h^3` kg

`(dM)/(dh)= 170\pi (3h^2)`

`\Rightarrow dM=510\pi h^2\ dh`

Work done = Force × displacement

= Mass × acceleration×displacement

Here acceleration= acceleration due to gravity = 9.8 m/s²

The displacement when the hot chocolate level is h is = (4-h)

Work done (dw)
`=(510\pi h^2 )(9.8)(4-h)dh`
`=4998\pi h^2 (4-h)dh`

Work done = W
`=\int_0^34998\pi h^2 (4-h)dh`

`=\int_0^34998\pi [4h^2-h^3]dh`

`=4998\pi [4(h^3)/(3)-(h^4)/(4)]_0^3`

`=4998\pi [(4(3^3)/(3)-(3^4)/(4))-(4(0^3)/(3)-(0^4)/(4))]`

=247401 J

The required work to empty the the tank is 247401 J.