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Thread: Conical paper cup - Work problem

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    Senior Member mollymcf2009's Avatar
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    Conical paper cup - Work problem

    A large conical paper cup has height 1 m. and top radius $\displaystyle \frac{1}{4} m.$ It is filled with water to a height of $\displaystyle \frac{3}{4} m$. Write (but do not evaluate) an integral that equals the work required to empty the cup by pumping all of the water to the top. (Note: The density of water is $\displaystyle 1000kg/m^3$. Answer needs to be in terms of a single variable.


    **I'm sure there is a good explanation in a previous thread that I could use. If someone knows of one and can direct me there, that's fine. Or if someone feels like walking me through this, I'd love it! I never really fully understood these work problems.
    Thanks
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    Quote Originally Posted by mollymcf2009 View Post
    A large conical paper cup has height 1 m. and top radius $\displaystyle \frac{1}{4} m.$ It is filled with water to a height of $\displaystyle \frac{3}{4} m$. Write (but do not evaluate) an integral that equals the work required to empty the cup by pumping all of the water to the top. (Note: The density of water is $\displaystyle 1000kg/m^3$. Answer needs to be in terms of a single variable.
    work = $\displaystyle \int WALT$

    $\displaystyle W$ = weight density
    $\displaystyle A$ = cross-sectional area of a representative horizontal slice of liquid in terms of y
    $\displaystyle L$ = lift distance of a horizontal slice in terms of y
    $\displaystyle T$ = slice thickness ... $\displaystyle dy$

    $\displaystyle W = \frac{mg}{V} = 9800 \, \frac{N}{m^3}$

    $\displaystyle A = \pi \left(\frac{y}{4}\right)^2$

    $\displaystyle L = 1 - y$

    $\displaystyle T = dy$

    work in N-m ...

    $\displaystyle \int_0^{\frac{3}{4}} 9800 \cdot \pi \left(\frac{y}{4}\right)^2 \cdot (1-y) \, dy$
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