Geometry

The total area of all the faces of a rectangular prism is 22, and the total length of all its edges is 24. Find the length of the internal diagonal of the prism.
Thank you!

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  1. let the sides be a, b, and c
    We know:
    4a +4b + 4c = 24 ----> a+b+c = 6
    and
    4ab + 4ac + 4bc = 22
    2ab + 2ac + 2bc = 11

    The length of the diagonal is
    √(a^2 + b^2 + c^2)

    by algebra, we know
    (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
    subbing in
    6^2 = a^2 + b^2 + c^2 +11
    a^2 + b^2 + c^2 = 25
    then ..

    √(a^2 + b^2 + c^2) = √25 = 5

    The diagonal is 5

    (nice problem)

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    posted by Reiny
  2. If the dimensions are x,y,z then we have

    2xy+2xz+2yz = 22
    4x+4y+4z = 24

    xy+xz+yz = 11
    x+y+z = 6

    How about 1,2,3
    diagonal is √14

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    posted by Steve
  3. Sorry, tried 5 but it didn't work. The hint said that it was a simplified square root?

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  4. How did I possible see 4 of equal sides ?

    but... my algebra worked out sooo nice!

    Reminds me of the classic StarTrek episode where Cpt Picard is tortured by the Kardashians and sees 4 lights.
    https://www.youtube.com/watch?v=moX3z2RJAV8

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    posted by Reiny
  5. Let the sides be a, b, and c. Then the total surface area is 2(ab+bc+ca)=22, and total length of all the edges is 4(a+b+c)=24. Hence, ab + ac + bc = 11 and a + b + c = 6.

    We want to find the length of the internal diagonal, which is √a^2 + b^2 + c^2. First, we square the equation a + b + c = 6, to get
    a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 36.

    Therefore,
    a^2 + b^2 + c^2 = 36 - 2(ab + ac + bc) = 36 - 2 * 11 = 14,
    which means that the length of the internal diagonal is √14.

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    posted by me

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