<p>I have a work problem that I can't solve:
A tank in the shape of a frustum with top radius 6 ft and bottom radius 3 ft is 8 ft high and full of water. How much work is required to pump all the water out? Use 62.5 lb/ft^3 as the weight of water.
The correct answer is approximately 1.04*10^5 ft-lbs, but can anyone show me how to get that?</p>
<p>Well, work is the integration of force over a distance x. Find out how much force you have using the weight of the water and the pressure (which is force/area).</p>
<p>There is a way to do it by finding the area of a slice and then using disk method to integrate all the slices together to find the volume. Work equals mgh and mass=density<em>volume, so work equals the integral of density</em>volume<em>area</em>dx.
But I’m still not getting the right answer.</p>
<p>What answer are you getting? </p>
<p>I never did work problems in Calculus, but from a physics perspective, you might want to try something along the lines of integration of density<em>volume</em>weight of water<em>height (which is weight</em>height, which should be your work).</p>
<p>Or, another thing you could try is to generate two equations (should be lines) based on the information you have. So make two lines so that their initial difference in y is 6 and their final difference in y is 3 after a change in x of 8. Then, use your disk method, find the volume of that and multiply that by weight of water.</p>
<p>I’ve never actually done one of these myself, so I’m kinda guessing based on what I learned in Physics and Calc last year.</p>
<p>You were on the right track</p>
<p>Work = mgh
m=rho<em>V
V=pi</em>r^2<em>h
or for a thin slice, V=pi</em>r^2*dh</p>
<p>so
integral of mgh = integral of rho<em>g</em>h<em>pi</em>r^2*dh
rho,g,pi are constants</p>
<p>What’s r as a function of h? Using h = 0 at the top (so that no work is required to pump fluid that is already at the top…)
r(h=0)=6
r(h=8)=3
gives</p>
<p>r=6-(3/8)*h</p>
<p>so
rho<em>g</em>pi<em>(Integral of h</em>(6-3/8*h)^2 dh from h=0 to h=8)</p>
<p>gives 103000 lb-ft</p>
<p>edit: g = 1 in these units.</p>
<p>Should have been more clear:</p>
<p>103000 lbf-ft (lbf = pound-force)</p>
<p>g = 1 when converting from lbm (pound-mass) to lbf</p>