Question

Your friend has a cube of gold. You have a cube of gold that weighs twice as much. Your cube is _______ as long as your friend's, and it has _________times the surface area. You can imagine how this would work with a spherical diamond, so we won't repeat that part here. Here is another question: The ratio of the weight of your cube and the surface area of your cube is ______times as large as the corresponding ratio of your friend's cube.

Answer 1

well, volume of a cube with edge length \(r\) is \[V=r^3\]
so doubling the edge length means volume (and thus mass or weight) goes up by a factor of \(2^3=8\)

that means that edge length must be \(
a r\) such that \((ar)^3=2r^3\) or \(a^3=2\) for your cube to be twice the weight as a cube with edge length \(r\). from that you should be able to work out the rest...