Question

Suppose you have two cubes, A and B. Cube A is composed of 216 smaller cubes \((6 \times 6 \times 6)\) and cube B is composed of 27 smaller cubes \((3 \times 3 \times 3)\). 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 216 surface cubes out of 216 total cubes, fraction 1. Cube B has 54 surface cubes out of 27 total cubes, fraction 2. Cube B has a higher fraction on the surface.

Step-by-step solution

Calculate Surface Cubes of Cube A

To calculate the number of small cubes on the surface of cube A, recognize that each face of the cube is one layer of small cubes. Since cube A is a 6x6x6 cube, each face is made up of 6x6=36 small cubes. Cube A has 6 faces, so the total number of cubes on the surface is 6 faces x 36 cubes per face = 216 surface cubes.

Calculate Total Cubes for Cube A

Cube A is composed of 216 smaller cubes in total since it is a 6x6x6 cube. This number is already given.

Calculate Surface Cubes of Cube B

Using the same method for Cube B, which is a 3x3x3 cube, each face is made up of 3x3=9 small cubes. Cube B also has 6 faces, thus the total number of cubes on the surface is 6 faces x 9 cubes per face = 54 surface cubes.

Calculate Total Cubes for Cube B

Similarly, Cube B is composed of 27 smaller cubes in total since it is a 3x3x3 cube.

Calculate the Fraction of Surface Cubes for Both Cubes

The fraction of small cubes on the surface for Cube A is the total number of surface cubes (216) divided by the total number of cubes (216) which is 216/216 = 1. The fraction for Cube B is 54 surface cubes divided by 27 total cubes which is 54/27 = 2.

Compare the Fractions

Since the fraction for Cube A is 1 and for Cube B is 2, Cube B has a higher fraction of its cubes on the surface.

Key concepts

Geometric Solids

Understanding geometric solids is fundamental when it comes to grasping various mathematical concepts. Geometric solids, or three-dimensional shapes, have length, width, and height, which distinguish them from two-dimensional figures that only have length and width. Common examples include cubes, spheres, and cylinders.

When discussing cubes, as in the given exercise, each side or face of the cube is a square, and all edges are of equal length. The cube's geometry is pivotal when calculating its surface area and volume, which are both essential in determining its surface area to volume ratio. This ratio has practical implications, such as in biology where it affects how quickly organisms can heat or cool.

Cube Surface Area Calculation

To calculate the surface area of a cube, you need to understand that it has six faces, each being a square of the same size. The surface area of one face is calculated by squaring the length of one of its sides. Since all sides are equal, if the side length is denoted by 's', then the area of one face is given by the formula:
\[ \text{Area of one face} = s^2 \].
To find the total surface area of the cube, you simply multiply the area of one face by the total number of faces, which is six:
\[ \text{Total surface area} = 6 \times (s^2) \].
This straightforward calculation helps us to understand the proportion of the cube's surface that is exposed relative to its overall volume.

Fraction Comparison

Comparing fractions is a fundamental skill in mathematics, which allows us to understand proportions and make decisions based on those proportions. To compare fractions effectively, you should consider their numerators and denominators. When the denominators are different, finding a common denominator or converting fractions to decimals may be necessary for an accurate comparison.

However, in some cases, as with the exercise given, the fractions may represent a part-to-whole relationship within a single geometric shape, such as the number of surface cubes to the total number of cubes in a larger cube. When comparing these kinds of fractions, considering the context of the problem, such as the size of each cube and the significance of the surface cubes, is crucial to interpreting the results correctly.