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    Topics || Problems

    If a cube has an edge equal to the diagonal of another cube, what is the ratio of their volumes?

    Let \(a\) be the side of the first cube and \(b\) be the diagonal of the second cube and \(v_a\) and \(v_b\) be their volumes respectively.

    \(v_a = a^3\)

    If x is the side of the second cube then,

    \(b^2 = (\sqrt{2} x)^2 +x^2\) \(b^2 = 3x^2\)
    \(x = \frac{b}{\sqrt{3}}\)
    \(v_b = x^3\)
    \(v_b = (\frac{b}{\sqrt{3}})^3\)
    \(v_b = \frac{b^3}{3^{\frac{3}{2}}}\)

    Thus the ratio between the volume of the two cubes is,

    \(\frac{v_a}{v_b} = \frac{a^3}{\frac{b^3}{3^{\frac{3}{2}}}}\)

    \(\frac{v_a}{v_b} = \frac{a^3}{\frac{a^3}{3^{\frac{3}{2}}}}\) , but \(a = b\)

    \(\frac{v_a}{v_b} = \frac{1}{\frac{1}{3^{\frac{3}{2}}}}\)

    \(\frac{v_a}{v_b} = 3^{\frac{3}{2}}\)