Three cubes of volume 1, 8 and 27 are glued together at their faces. What is the smallest possible surface area of the resulting configuration? With full solution
To find the smallest possible surface area of the resulting configuration, we need to consider how the cubes are glued together.
Let's label the cubes A, B, and C according to their volumes: A for 1, B for 8, and C for 27.
To minimize the surface area, we want to arrange the cubes in a way that minimizes the number of faces exposed.
We can visualize three possible configurations based on how the cubes are glued:
Configuration 1:
In this configuration, the cubes are stacked vertically, with cube B on top of cube A and cube C at the bottom. This arrangement results in 6 faces exposed: the top, the bottom, and four vertical faces. Since the cubes have volumes 1, 8, and 27, the side lengths of the respective cubes are 1, 2, and 3. Therefore, the total surface area of this configuration is (1x1) + (1x1) + (2x1) + (2x1) + (1x2) + (1x2) + (3x1) + (3x1) = 20 units.
Configuration 2:
In this configuration, cube C forms the base, cube B is glued to the top face of C, and cube A is glued to the top face of B. This arrangement results in 5 faces exposed: the four vertical faces and the top face. The lateral surface area of cube C is 4 x (3x3) = 36 units, since the side length of cube C is 3. The lateral surface area of cube B is 4 x (2x2) = 16 units, and the lateral surface area of cube A is 4 x (1x1) = 4 units. Adding these up, we get a total surface area of 36 + 16 + 4 = 56 units.
Configuration 3:
In this configuration, cube C forms the base, cube B is glued to one of the vertical faces of cube C, and cube A is glued to the remaining vertical face of C. This arrangement also results in 5 faces exposed: the four vertical faces and the top face. Similarly as in Configuration 2, we find a total surface area of 56 units.
Comparing the surface areas of the three configurations, we see that Configuration 1 has the smallest surface area, with a total of 20 units. Therefore, the smallest possible surface area of the resulting configuration is 20 units.