Final answer:
To quadruple the density of a cube while keeping its mass and shape the same, each side length of the cube must be halved from its original length.
Step-by-step explanation:
To determine the new length of each side of a cube when the density is quadrupled while keeping the mass and shape the same, you first need to understand the relationship between density (ρ), mass (m), and volume (V). Density is defined as mass divided by volume (ρ = m/V), and for a cube, the volume is the cube of the side length (V = l³). To quadruple the density with the same mass, you must reduce the volume to one-fourth of its original value since ρ is inversely proportional to V when m is constant.
Therefore, each side of the cube must be halved (l/2) to achieve the fourfold increase in density. If l is the original length, then the new length l' would be l' = l/√4 = l/2 for each side of the cube to maintain four times the original density with the same mass and shape.