Question

A tank has the shape of an inverted circular cone with height 16m and base radius 3m. The tank is filled with water to a height of 9m. Find the work required to empty the tank by pumping all of the water over the top of the tank. Use the fact that acceleration due to gravity is 9.8 m/sec2 and the density of water is 1000kg/m3. Round your answer to the nearest kilojoule.

Answer 1

`W=17085KJ`

Step-by-step explanation:

From the question we are told that:

Height
`H=16m`

Radius
`R=3`

Height of water
`H_w=9m`

Gravity
`g=9.8m/s`

Density of water
`\rho=1000kg/m^3`

Generally the equation for Volume of water is mathematically given by

`dv=\pi*r^2dy`

`dv=(\piR^2)/(H^2)(H-y)^2dy`

Where

y is a random height taken to define dv

Generally the equation for Work done to pump water is mathematically given by

`dw=(pdv)g (H-y)`

Substituting dv

`dw=(p(=(\piR^2)/(H^2)(H-y)^2dy))g (H-y)`

`dw=(\rho*g*R^2)/(H^2)(H-y)^3dy`

Therefore

`W=\int dw`

`W=\int((\rho*g*R^2)/(H^2)(H-y)^3)dy`

`W=\rho*g*R^2}{H^2}\int((H-y)^3)dy)`

`W=(1000*9.8*3.142*3^2)/(9^2)[((9-y)^3)}^9_0`

`W=3420.84*0.25[2401-65536]`

`W=17084965.5J`

`W=17085KJ`

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