Question

Suppose you have two cubes, A and B. Cube A is composed of 512 smaller cubes \((8 \times 8 \times 8)\) and cube B is composed of 64 smaller cubes \((4 \times 4 \times 4)\). Calculate the fraction of small cubes on the surface of cubes A and B. Which cube has a higher fraction at the surface?

Short answer

Cube A has \(\frac{6 \times 64 - 12 \times 6 - 8}{8 \times 8 \times 8} = \frac{344}{512} = \frac{43}{64} \) of its cubes on the surface. Cube B has \(\frac{6 \times 16 - 12 \times 2 - 8}{4 \times 4 \times 4} = \frac{56}{64} = \frac{7}{8}\) of its cubes on the surface. Cube B has a higher fraction at the surface.

Step-by-step solution

- Calculate the Number of Smaller Cubes on the Surface of Cube A

Cube A has 6 faces, and each face is made up of \(8 \times 8 = 64\) smaller cubes. Since each edge cube is shared by two faces and each corner cube is shared by three faces, subtract the edge cubes once and the corner cubes twice to avoid double-counting. There are 12 edges with 6 cubes each (after subtracting the corners) and 8 corner cubes, leading to \(6 \times 64 - 12 \times 6 - 8\) surface cubes.

- Calculate the Number of Smaller Cubes on the Surface of Cube B

Similarly, Cube B has 6 faces, each composed of \(4 \times 4 = 16\) smaller cubes. There are 12 edges with 2 cubes each (after subtracting the corners) and 8 corner cubes. Repeat the subtraction to avoid double-counting, yielding \(6 \times 16 - 12 \times 2 - 8\) surface cubes.

- Calculate the Surfaced Fraction for Both Cubes

Divide the number of surface cubes by the total number of cubes for each cube: \(SurfaceFractionA = \frac{6 \times 64 - 12 \times 6 - 8}{8 \times 8 \times 8}\) and \(SurfaceFractionB = \frac{6 \times 16 - 12 \times 2 - 8}{4 \times 4 \times 4}\). Simplify these fractions to get the surfaced fraction for both Cube A and Cube B.

- Compare the Surfaced Fractions

After simplifying the fractions from Step 3, compare the surfaced fractions of Cube A and Cube B to determine which cube has a higher fraction of cubes at the surface.

Key concepts

Geometric Fraction Calculation

When faced with the task of finding the fraction of smaller cubes at the surface of two different-sized larger cubes, we delve into geometric fraction calculation. This approach involves comparing parts of a geometric shape, like the surface cubes, to the whole, such as the entire cube made up of smaller units.

In the exercise, for both Cubes A and B, calculating the number of surface cubes requires understanding the cube's geometry. By multiplying the number of cubes along one edge squared by six (since a cube has six faces), we find the number of smaller cubes that initially appear to create the surface. However, to compute the accurate fraction of surface cubes, we must account for the cubes shared between multiple faces to avoid counting them more than once.

For the edge cubes, shared between two faces, each is counted only once. Corners form the intersection of three faces, and thus each of the corners must be counted less to adjust for the double-counting. After we adjust for these overlaps, the geometric fraction is found by dividing the corrected number of surface cubes by the total number of cubes in the structure.

Surface Cubes of a Geometry

Examining the surface cubes of a geometry, particularly in cubic structures, highlights the unique properties of edges and corners. In a cube, edge cubes form a line between two faces, while corner cubes exist where three faces meet. For Cube A, with its edge consisting of 8 smaller cubes, excluding the corners, there are six cubes that form the inner part of the edge. Cube B, with a smaller edge size, has two such cubes per edge.

To calculate the number of surface cubes without over-counting, we subtract these edges and corners. For Cube A, we have 12 edges with 6 cubes each. Similarly, for Cube B, there are 12 edges, but only 2 cubes each, after corner subtraction. The correction ensures that every cube is represented accurately within our surface calculation.

Accounting for Overlaps

The presence of edges and corners shows us that geometry is not just about shapes, but also about understanding their interconnections and how those connections affect our calculations.

Volume of Cubic Structures

Finally, let's explore the volume of cubic structures. The volume of a cube is found by raising the edge length to the power of three. For a standard cube, this calculation is straightforward; however, when the cube is an assembly of smaller units, we multiply the number of these smaller cubes along the edge by themselves three times.

In Cube A, the edge contains 8 smaller cubes, so the volume is \(8 \times 8 \times 8\), whereas Cube B, with a 4-cube edge, has a volume of \(4 \times 4 \times 4\). When we consider the volume in relation to the surface area, as we did earlier with surface fractions, we gain a deeper understanding of the 'surface area to volume ratio,' a significant concept in many scientific fields, including biology, where it can affect rates of heat loss and nutrient intake.