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Math Help - [SOLVED] latitude

  1. #1
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    [SOLVED] latitude

    How would i show the following:

    Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).
    Last edited by nerdo; November 28th 2009 at 09:19 AM.
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  2. #2
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    How would i show the following:

    Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).
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  3. #3
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    Quote Originally Posted by nerdo View Post
    How would i show the following:

    Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).
    Do an internet search for "SPHERICAL CAP"

    or look here:Spherical Cap -- from Wolfram MathWorld

    Since you are looking only for a per cent value, you do not require the radius of the earth in miles or kilometers, just use "R" for the radius.

    Calculate the total surface area of a sphere,
    then the area of the spherical cap.

     \dfrac{\text{AreaSphericalCap}}{\text{SurfaceAreaS  phere} }100 = \text{percentage}

    .
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  4. #4
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    Quote Originally Posted by aidan View Post
    Do an internet search for "SPHERICAL CAP"

    or look here:Spherical Cap -- from Wolfram MathWorld

    Since you are looking only for a per cent value, you do not require the radius of the earth in miles or kilometers, just use "R" for the radius.

    Calculate the total surface area of a sphere,
    then the area of the spherical cap.

     \dfrac{\text{AreaSphericalCap}}{\text{SurfaceAreaS  phere} }100 = \text{percentage}

    .

    Thanks for the help, but i would like to ask how would you calculate the radius.
    Last edited by nerdo; November 29th 2009 at 09:15 AM.
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  5. #5
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    Quote Originally Posted by nerdo View Post
    Thanks for the help, but i would like to ask how would you calculate the radius.
    Surface Area of a sphere:
    S_{\text{sphere}} = 4 \, \pi \, r^2

    Surface Area of a spherical cap
    S_{\text{cap}} = 2 \, \pi \, r \, h


    If you sliced the earth along the latitude 66.5 degress,
    the distance from the plane ( the flat part )
    to the north point is the distance h.

     h = R - R \sin (latitude)
     h = R(1 - \sin (latitude))

    replacing h (in the spherical cap equation above)
    S_{\text{cap}} = 2 \, \pi \, r^2 (1-sin(latitude))

    the ratio:
     \dfrac{ \text{SphericalCapArea}}{\text{SurfaceAreaOfSphere  }}


    \dfrac{2\,\pi\,r^2(1-\sin(latitude))}{4\,\pi\,r^2}

    Radius cancels.
    Pi cancels.

    .
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  6. #6
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    Quote Originally Posted by aidan View Post
    Surface Area of a sphere:
    S_{\text{sphere}} = 4 \, \pi \, r^2

    Surface Area of a spherical cap
    S_{\text{cap}} = 2 \, \pi \, r \, h


    If you sliced the earth along the latitude 66.5 degress,
    the distance from the plane ( the flat part )
    to the north point is the distance h.

     h = R - R \sin (latitude)
     h = R(1 - \sin (latitude))

    replacing h (in the spherical cap equation above)
    S_{\text{cap}} = 2 \, \pi \, r^2 (1-sin(latitude))

    the ratio:
     \dfrac{ \text{SphericalCapArea}}{\text{SurfaceAreaOfSphere  }}


    \dfrac{2\,\pi\,r^2(1-\sin(latitude))}{4\,\pi\,r^2}

    Radius cancels.
    Pi cancels.

    .
    Thank you for the help,
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