A solid metal cuboid measuring \(6\) cm by \(5\) cm by \(4\) cm is melted down and made into a single cube. Calculate the length of an edge of the cube. Giv...

Melting and recasting conserves volume: the metal is rearranged but none is lost, so the cube must have exactly the same volume as the original cuboid. Finding the edge of the cube then means taking a cube root.

1. Volume of the cuboid: \(6 \times 5 \times 4 = 120\) cm\(^3\). Since the cube has the same volume, its edge \(x\) satisfies \(x^3 = 120\), so \[ x = \sqrt[3]{6 \times 5 \times 4} = \sqrt[3]{120} \] [M1]
2. Evaluating, \(\sqrt[3]{120} = 4.9324\ldots\), so the edge is \(4.93\) cm correct to 3 significant figures [A1]. Values of \(4.932\) to \(4.933\) are accepted.

Surface area is not conserved when a solid is melted down, only volume, so nothing about the \(6\) cm, \(5\) cm and \(4\) cm faces carries over to the cube other than their product. A tempting but wrong shortcut is to average the three dimensions, giving \(5\) cm; that is close here but is not a valid method, since \(5^3 = 125\) is not \(120\). The answer is sensible because \(4.93\) lies between the smallest and largest edges of the original cuboid.