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## Homework Statement

A cylindrical barrel, standing upright on its circular end, contains muddy water. The top of barrel, which has diameter 1 meter, is open. Height of barrel is 1.8 meter and it is filled to a depth of 1.5 meter. Density of water at a depth of h meters below surface is given by [tex]\delta[/tex](h)=1+kh kg/m

^{3}, where k is a positive constant. Find total work done to pump muddy water to top rim of barrel. Answer is .366(k+1.077)g[tex]\pi[/tex] joules

## Homework Equations

force of gravity on slice=density*g*volume

## The Attempt at a Solution

volume of slice[tex]\approx[/tex][tex]\pi[/tex](1/2)

^{2}[tex]\Delta[/tex]h m

^{3}

1+kh(g)([tex]\pi[/tex]/4)[tex]\Delta[/tex]h nt

work done on slice[tex]\approx[/tex]force*distance=1+kh(g)([tex]\pi[/tex]/4)(1.8-h)[tex]\Delta[/tex]h joules

total work=[tex]\int[/tex]

_{0}

^{1.5}[tex]\pi[/tex]/4g(1+kh)(1.8-h)dh joules=[tex]\pi[/tex]/4g(1.8h-h

^{2}/2+1.8kh

^{2}/2-kh

^{3}/3)