Three cubes of volume 1cm3, 8 cm 3 and 27cm3 are glued together at their faces. find the smallest possible surface area of the resulting configuration
You have cubes of sides
1 cm, 2 cm, and 3 cm
What do I see of the 3 cm cube?
I see 6 faces minus the part of the top face obstructed by the 2 cm cube,
so : 6(3^2) - 2^2 = 50 cm^2
what do I see of the 2 cm cube?
5 sides - part of the top obstructed by the 1 cm cube
= 5(2^2) - 1^2 = 19 cm
and of course I see 5 sides of the top cube, so the total
= 50 + 19 + 5 cm^2
= 74 cm^2
Well, when it comes to gluing things together, it can get a bit sticky! But let's dive into the problem.
To find the smallest possible surface area of the resulting configuration, we need to minimize the amount of surface area that is exposed. Since the cubes are glued together at their faces, we want to make sure that as many faces as possible are shared between adjacent cubes.
To achieve this, we can arrange the three cubes in a line, with the smallest cube at the center, and the other two cubes on either side.
This way, the smallest cube shares three of its faces with the two larger cubes, and the two larger cubes share one of their faces with each other. So, there will be a total of 9 faces exposed.
Therefore, the smallest possible surface area of the resulting configuration would be 9 square centimeters.
But, you know what they say, sometimes sticking things together can bring out the best out of them!
To find the smallest possible surface area of the resulting configuration, we need to arrange the three cubes in a way that minimizes the total surface area.
Let's denote the edge lengths of the three cubes as a, b, and c, respectively.
Given that the volume of the first cube is 1 cm³, we can express the edge length a as: a³ = 1 cm³, which gives us a = 1 cm.
Similarly, we can express the edge length b and c as: b³ = 8 cm³ and c³ = 27 cm³.
To minimize the total surface area, we need to arrange the cubes in a way that eliminates duplicate faces.
First, let's arrange the cubes in a straight line, with the cube of volume 1 cm³ at one end, the cube of volume 8 cm³ in the middle, and the cube of volume 27 cm³ at the other end.
In this configuration, the total surface area is:
Surface Area = (6 × a²) + (4 × b²) + (2 × c²)
Substituting the values, we have:
Surface Area = (6 × 1²) + (4 × 2²) + (2 × 3²)
Surface Area = 6 + 16 + 18
Surface Area = 40 cm²
Therefore, the smallest possible surface area of the resulting configuration is 40 cm².
To find the smallest possible surface area of the resulting configuration, we need to consider the different ways these cubes can be arranged when glued together.
Let's start by visualizing the cubes. We have three cubes with volumes 1 cm^3, 8 cm^3, and 27 cm^3. Let's label these cubes A, B, and C respectively:
Cube A: 1 cm^3
Cube B: 8 cm^3
Cube C: 27 cm^3
We can arrange these cubes in various ways by gluing their faces together. Let's explore the different possibilities:
1. Option 1: Cube A on top, Cube B in the middle (side by side with Cube A), and Cube C at the bottom (side by side with Cubes A and B).
This arrangement forms a vertical stack of cubes. The total surface area can be calculated as follows:
- Cube A: Since we are only interested in the smallest possible surface area, we consider the faces that are not glued to any other cube. Cube A has a total of five visible faces (one bottom face and four side faces).
- Cube B: Similar to Cube A, we consider the visible faces that are not glued to any other cube. Cube B has only four visible faces, as its top face is glued to Cube A and its bottom face is glued to Cube C.
- Cube C: Again, considering only the visible faces not glued to any other cube, Cube C has three visible faces (one top face and two side faces).
Adding up the visible faces of each cube:
Cube A: 5 faces
Cube B: 4 faces
Cube C: 3 faces
Total surface area = 5 + 4 + 3 = 12 cm^2
2. Option 2: Cube A on top, Cube B on the left side (adjacent to Cube A), and Cube C on the right side (adjacent to Cube A).
This arrangement forms a horizontal stack of cubes. The total surface area can be calculated as follows:
- Cube A: As before, we consider the visible faces not glued to any other cube. Cube A has six visible faces (one bottom face and five side faces).
- Cube B: For Cube B, we only consider the visible faces not glued to any other cube. Cube B has four visible faces.
- Cube C: Considering only the visible faces not glued to any other cube, Cube C has four visible faces.
Adding up the visible faces of each cube:
Cube A: 6 faces
Cube B: 4 faces
Cube C: 4 faces
Total surface area = 6 + 4 + 4 = 14 cm^2
3. Option 3: Cube A on top, Cube B on the left side (adjacent to Cube A), and Cube C on the bottom (adjacent to Cube A and B).
This arrangement also forms a horizontal stack of cubes. The total surface area can be calculated as follows:
- Cube A: As before, we consider the visible faces not glued to any other cube. Cube A has six visible faces.
- Cube B: For Cube B, we only consider the visible faces not glued to any other cube. Cube B has four visible faces.
- Cube C: Considering only the visible faces not glued to any other cube, Cube C has six visible faces.
Adding up the visible faces of each cube:
Cube A: 6 faces
Cube B: 4 faces
Cube C: 6 faces
Total surface area = 6 + 4 + 6 = 16 cm^2
From the three arrangements, we see that the smallest possible surface area of the resulting configuration is 12 cm^2 (Option 1: vertical stack of cubes).