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Math Help - Need Help with a Problem!

  1. #1
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    Exclamation Need Help with a Problem!

    I have a problem for Calculus II....I would greatly appreciate any help.


    Suppose that a pyramid has a square base of side 6meters and an altitude of 10 meters. In addition, the density of the pyramid at a point x meters from the vertex is proportional to x (and is given in kg/m^3). In terms of the proportionality constant, what is the total mass of the pyramid?
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  2. #2
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    Let b be the side of the pyramid base (6 metres) and h be the height of the pyramid (10 metres).

    Any horizontal slice through the pyramid will produce a square with side
    b(h-z)/h where z is the height above the bottom.

    The density is proportional to the distance x from the apex so:-
    density = a times x but note that x = h-z
    density= a(h-z)

    so the mass of a square at a height z above the bottom is its area (side times side) times the density at that level times dz

    dm = [b(h-z)/h]**2 times a(h-z) times dz

    Mass is integral from z=0 to z=h of

    (a*b**2/h**2) (h-z)**3 dz

    Which I assume you can do
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