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Thread: Cone and Hemisphere

  1. #1
    Member GAdams's Avatar
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    Cone and Hemisphere

    (a) pi * r * l = 1/2 (4 * pi * r^2)

    l = 2r

    (b) I can't do

    (c) I can't do

    Thanks
    Attached Thumbnails Attached Thumbnails Cone and Hemisphere-cone-hemisphere-i.jpg   Cone and Hemisphere-cone-hemisphere-ii.jpg   Cone and Hemisphere-cone-hemisphere-iii.jpg  
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  2. #2
    Bar0n janvdl's Avatar
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    For b)

    Use Pyhtagoras.

    $\displaystyle h^2 = l^2 + r^2 $

    $\displaystyle h = \sqrt{l^2 + r^2} $
    Last edited by janvdl; Jul 8th 2007 at 03:29 AM.
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  3. #3
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    Hello, GAdams!

    (b) Find the perpendicular height, $\displaystyle h$, of the cone in terms of $\displaystyle r$.
    From the diagram in (a), we can use Pythagorus (as janvdl suggested):

    . . $\displaystyle h^2 + r^2\:=\:L^2\quad\Rightarrow\quad h \:=\:\sqrt{L^2 - r^2}$

    Since $\displaystyle L = 2r$, we have: .$\displaystyle h \:=\:\sqrt{(2r)^2 - r^2} \:=\:\sqrt{3r^2}\quad\Rightarrow\quad\boxed{h \:=\:\sqrt{3}r}$




    (c) Find the ratio of the volumes of the cone and the hemisphere.
    The volume of a cone is: .$\displaystyle V \:=\:\frac{1}{3}\pi r^2h \:=\:\frac{1}{3}\pi r^2(\sqrt{3}r) \:=\:\frac{\sqrt{3}}{3}\pi r^3$

    The volume of a hemisphere is: .$\displaystyle \frac{1}{2} \times \frac{4}{3}\pi r^3\:=\:\frac{2}{3}\pi r^3$

    The ratio is: .$\displaystyle \frac{V_c}{V_h} \;=\;\frac{\frac{\sqrt{3}}{3}\pi r^3}{\frac{2}{3}\pi r^3} \:=\:\boxed{\frac{\sqrt{3}}{2}}$

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  4. #4
    Member GAdams's Avatar
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    The volume of a cone is confusing me:

    I can't see how you got from 1/3 * pi * sq.rt3r to sq.rt3/3 * pi * r^3
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  5. #5
    Bar0n janvdl's Avatar
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    I tried googling for the formulas in a), but i struggled to find them. So i couldn't prove that formula. I think i struggled to find the hemisphere one.

    So if $\displaystyle l = 2r $ then just plug that in like Soroban did.
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