Imagine that you have gelatin cut into three cubes: the side...

Imagine that you have gelatin cut into three cubes: the side of cube A is $a$ cm long, the side of cube $B$ is double the side of $A$, and the side of cube $C$ is three times the side of A. Compare the following properties of the cubes: (a) density, (b) volume, (c) surface area, (d) cross-sectional area, and (e) mass.

Imagine that you have gelatin cut into three cubes: the side of cube A is $a$ cm long, the side of cube $B$ is double the side of $A$, and the side of cube $C$ is three times the side of A. Compare the following properties of the cubes:
(a) density, (b) volume,
(c) surface area,
(d) cross-sectional area, and
(e) mass.

Key Concepts

Density

Density is defined as mass per unit volume, and for a homogeneous material like gelatin, it remains constant regardless of the object's shape or size. When comparing cubes of different sizes made from the same material, the density does not change, as it is an intrinsic property of the material rather than of its geometric dimensions.

Volume

The volume of a cube is calculated by raising its side length to the third power (side^3). This cubic relationship means that if the side length of a cube is scaled by a factor, the volume increases by the cube of that factor. This concept is essential in understanding how changes in dimensions affect the overall capacity or mass of the object when the density is constant.

Surface Area

The surface area of a cube is determined by the formula 6 multiplied by the square of its side length (6*side^2). This quadratic relationship between the side length and surface area means that as the cube’s dimensions are scaled up, the surface area increases by the square of the scaling factor, which is significantly less rapid than the increase in volume.

Cross-sectional Area

The cross-sectional area of a cube, or the area of one face, is given by the square of its side length. This concept uses the idea of scaling in two dimensions, meaning that this area scales with the square of the side length. It is directly related to the surface area calculation and provides insight into how two-dimensional aspects of an object change with size.

Mass

Mass is directly proportional to volume when the density of the material is constant. Since mass is computed as the product of density and volume, any scaling in the volume directly results in a proportional change in mass. This directly connects the geometric property of volume with the physical property of mass.

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