Consider eight two-cubic centimeter (2 cm3) sugar cubes stacked so that they form a single 2 x 2 x 2 cube. How does the surface area of the single, large cube compare to the total surface area of the individual eight cubes? Report your answer as a ratio. Be sure to show all calculations leading to an answer.
To find the surface area of the single, large cube, we need to calculate the sum of the surface areas of all six sides.
Each side of the small cubes has an area of 2 cm^2, as each face is a square with sides measuring 1 cm.
Since there are 8 small cubes, the total surface area of all the small cubes is 8 * 6 * 2 cm^2 = 96 cm^2.
The single, large cube has six faces, each with an area of 4 cm^2 since each side measures 2 cm.
Therefore, the total surface area of the single, large cube is 6 * 4 cm^2 = 24 cm^2.
So, the ratio of the surface area of the single, large cube to the total surface area of the individual eight cubes is 24 cm^2 : 96 cm^2.
Simplifying this ratio, we have 1:4.
Therefore, the surface area of the single, large cube is one-fourth the total surface area of the individual eight cubes.
To compare the surface areas of the single, large cube and the individual eight sugar cubes, we need to calculate the total surface area for each and then find the ratio.
First, let's calculate the surface area of the large cube. Since it is a 2 x 2 x 2 cube, each face has an area of 2 x 2 = 4 square centimeters. There are six faces in total, so the total surface area of the large cube is 6 x 4 = 24 square centimeters.
Next, let's calculate the surface area of each smaller sugar cube. Since each sugar cube has dimensions of 1 x 1 x 1, each face has an area of 1 x 1 = 1 square centimeter. There are six faces in total, so the total surface area of one sugar cube is 6 x 1 = 6 square centimeters.
Since we have eight sugar cubes, the total surface area of all the individual cubes is 8 x 6 = 48 square centimeters.
To compare the surface areas, we will find the ratio of the surface area of the large cube to the total surface area of the individual eight cubes: 24/48 = 1/2.
Therefore, the ratio of the surface area of the single, large cube to the total surface area of the individual eight cubes is 1:2.