Sphere Problem - Please correct

  1. 10 000 cubic km of water can cover all the continents with a layer 7 cm deep.

    Find the depth of the water layer if 10 000 cubic km of water is spread over a sphere with radius R = 6370 km.

    My calculations:

    Volume of the sphere: 4/3*π*r^3

    R* = R + d (d = depht of the water=> searched)
    d = R* - R
    V* = V + v (V = 4/3*π*6370^3 ; is the little v equal to the 10 000 km^3?)

    Would that be correct:

    V = 4/3*π*6370^3 = 1.0826969e+12 km^3 (?)
    v = 10 000 km^3 (?)

    Total V*: 1.0826969e+12

    Next we search R*:
    Volume of the sphere: V* = 4/3*π*R*^3
    => R*^3 = V*/π*3/4
    = (1.0826969e+12/π)*3/4 => take the third root = 1.73205080757

    So d ~ 1.73 cm ?


    Please correct this! I'm really not sure if it's correct.

    Can someone please help and explain what I did wrong?
  2. jcsd
  3. mfb

    Staff: Mentor

    How did you get that number (it is wrong)? I don't understand your calculations. Where do you compare R* with R and where do the "cm" for the final answer come from?

    Your approach will get a problem with the accuracy, as you subtract numbers extremely close together. The depth of the water will be tiny compared to the radius of the sphere, it is easier to divide the volume by the surface area of the sphere.

  4. Volume of the sphere: 4/3*π*r^3 ---> is 10 000 km^3 the volume?
    Surface of the sphere: 4*π*r^2 ---> it's 4/3*π*6370^3 = 509.9 * 10^6 , isn't it?

    So I wanted to devide 10 000 km^3 by 509.9 * 10^6, but it doesn't funcion.

    What did I wrong?
  5. WWGD

    WWGD 1,798
    Science Advisor
    Gold Member

    You mention in the OP that the water can cover the continents; not the surface of the earth itself. So maybe you should divide by the surface area of the continents --around 150 million square kilometers -- and not by the surface area of the whole earth.
  6. You mean, 10 000 km^3 / 150 mio km^2
    But I can't divide something smaller by something bigger, or can I? I'm confused...
  7. WWGD

    WWGD 1,798
    Science Advisor
    Gold Member

    Why not? Then transform the result into cm. units.
  8. That gives me 66.6666666667 cm.

    But the solution should be around 2 cm. So something must be wrong with that equation, but what... ?
  9. WWGD

    WWGD 1,798
    Science Advisor
    Gold Member

    Sorry I was thinking of the first problem; about the earth, not the second one.
  10. Surface area sphere = 4πr2
    in this case 4 X π X 6370 X6370
    Divide 100000 by the answer to get the depth as a fraction of a kilometre. Multiply by 100000 to convert to centimetres - I get 1.96....
  11. SteamKing

    SteamKing 10,972
    Staff Emeritus
    Science Advisor
    Homework Helper

    I thought we covered this in basic arithmetic.

    What is 1/2? Isn't 2 bigger than 1? Cause, you know, fractions.
  12. WWGD

    WWGD 1,798
    Science Advisor
    Gold Member

    Compare this with the results you just got.
  13. mfb

    Staff: Mentor

    Use the 509.9 * 10^6 km^2 you calculated instead of the 150 * 10^6 km^2, and be careful with the conversion factor from kilometers to centimeters, then you should get the right result.
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