If a cube is built from an one inch cubes and if each new cube has an edge N times as long as the previous model what would be true of consecutive surface areas
To determine the relationship between consecutive surface areas of cubes with increasing edge lengths, we need to understand how the surface area of a cube changes with respect to its edge length.
The surface area of a cube is given by the formula: A = 6s^2, where A represents the surface area and s represents the length of each side of the cube.
In this case, we have a sequence of cubes where each new cube has an edge N times as long as the previous one. Let's represent the initial cube's edge length as "x". Therefore, the edge length of the second cube would be "Nx", the third cube would have an edge length of "N^2x", the fourth would be "N^3x", and so on.
Now, let's find the surface area of each cube in terms of "x" and "N". For the initial cube, the surface area is A_1 = 6x^2. For the second cube, the surface area is A_2 = 6(Nx)^2 = 6N^2x^2. For the third cube, it is A_3 = 6(N^2x)^2 = 6N^4x^2.
From this pattern, we can observe that the surface area of each consecutive cube is a multiple of the surface area of the previous cube. In general, the surface area of the nth cube would be A_n = 6N^(2n-2)x^2.
In conclusion, the consecutive surface areas of the cubes, in terms of their edge lengths, follow a pattern where each surface area is a multiple of the previous one.