How Is Work Calculated When Pumping Muddy Water from a Barrel?

[eestep]

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/m³, 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 m³
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-h²/2+1.8kh²/2-kh³/3)

 

[Mark44]

Here are some tips on LaTeX.

Use one pair of tags for a whole line, rather than piecemeal.
Inside LaTeX tags, use _n for subscripts and ^n for exponents. If the subscript or exponent is more than a single character, surround them with braces - {}
Integrals look like this (without the leading spaces in the brackets):
[ tex]\int_{a}^{b} f(h) dh [ /tex]