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Density, Volume and Slicing Problem

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data
    A compressible liquid has density which varies with height. AT the level of h meters above the bottom, the density is 40(5 - h) kg/m3
    a) The liquid is put in the containers below. The cross sections of the container are isosceles triangles. It has straight sides and looks like a triangular prism. How many kg will it hold when placed as shown on the left, resting on one triangular side?​
    DensityPRoblem.jpg
    ^There is a picture of the object.

    2. Relevant equations
    Mass = Volume times Density



    3. The attempt at a solution
    My theory is that since M = V*D, and you're given the density, should I slice and solve for volume, then evaluate the integral of volume and the given integral for density then multiply the results to get the mass?
    [tex]\int^{4}_{0}2.5 dh[/tex] and [tex]40\int^{4}_{0}5-h dh[/tex]

    I got to the first integral by taking the volume of the first slice (triangular prism):
    [tex]\sum\frac{1}{2}b*l*\Delta h[/tex] [tex]\rightarrow[/tex] [tex]\sum\frac{1}{2}2*2.5\Delta h[/tex] [tex]\rightarrow[/tex] lim as [tex]\Delta h[/tex] [tex]\rightharpoonup[/tex][tex]\int^{4}_{0}2.5 dh[/tex]

    To find the mass, according to the equation M=VD, should I solve both integrals , one being V and one being D, then multiply?
     
  2. jcsd
  3. Mar 3, 2010 #2
    Ah... It's starting to make more sense now, thanks. So since density is varying with the height we make it all one integral rather than separate ones. Part B is the same question except with the 4m on the ground on its point with the rectangle side up. I figured once I'd managed A, B would be simple. Thanks!
     
  4. Mar 3, 2010 #3

    cronxeh

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    Last edited by a moderator: Apr 24, 2017
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