June 13th, 2010

Question of the Week #2

Although I would like the QotW to be more applied I sometimes consider interesting proportions between geometric shapes. Imaging you have two perfect spherical oranges on a table. They are touching and there is a compressed Eiffel Tower shape between them and the table. You have a bowl of grapes and would like to roll the biggest one possible through this hole. In terms of the size of the oranges, what is the largest grape that will pass through the whole without pushing apart the oranges?

An assumption:

  • The radius of the orange is R
  • My Solution
    The solution relies on defining a triangle of known lengths. We can define the following lengths in terms of R and r, the radius of the grape.

  • The distance from the orange centre to the grape centre is R + r, this is the hypotenuse
  • The distance from the orange centre horizontally across to above the grape centre is simply R
  • The vertical distance from the centre of the grape to the point where both oranges touch is R – r
  • We now have the three sides of the triangle that we can drop into Pythagoras’ equation.

    h² = x² + y²

    Using our variables:

    (R+r)² = R² + (R-r)²

    Let’s explode, clean up, then isolate for r:

    R² + r² + 2Rr = R² + R² + r² – 2Rr

    Resulting in:

    r = R/4

    The grape must be a quarter the radius of the orange in order to just squeeze through the hole.

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