Circular Reasoning

[ThisIsMyName]

I was given this problem and I think I've got about half way through but now I'm totally stumped. Here's the problem:

 

 

The point is to find the volume. The 3, 2, & 1 are ratios, not the real number; but the 7√2 is real (it comes out to be about 9.8). What I did was take a triangle out using 7√2 as one side, 2 and another, and then labeling the remaining side d for diagonol. After a lot of scribling and trying various formulas, I have found myself here. If anyone can point me in the right direction or walk through it with me, I'd be extremely grateful. :eek:

[HydrogenBond]

The diagonal squared is equal to the sum of the squares of the three x,y,z, sides of the box. Since you know the side ratios, the three sides are x=x,y=2x,z=3x, Solve for x, then y, and then z. Then multipy x,y,z, to get the volume.

[Mohit Pandey]

Hello to all!

I have another problem. Solve it with explanation.

 

The six faces of a cube are painted in a manner that no two adjacent sides have the same colour. The three colours used in painting are red, blue and green. The cube is then cut into 36 smaller cubes in a manner that 32 cubes are of one size and the rest of a bigger size and each of the bigger cubes has no red side. Answer the following questions based on this.

1. How many cubes in all have a red side?

2 How many cubes in all have only one coloured side?

3.How many cubes are coloured on three sides ?

4. How many cubes are there which have two or more sides painted?

5.How many cubes are there which are painted on two sides only?

[Qfwfq]

The six faces of a cube are painted in a manner that no two adjacent sides have the same colour. The three colours used in painting are red, blue and green. The cube is then cut into 36 smaller cubes in a manner that 32 cubes are of one size and the rest of a bigger size and each of the bigger cubes has no red side.

The first thing to deduce is trivial, that there are 4 bigger cubes, and question 3 is easy because it's 1 for each vertex. The next thing that can be deduced, fairly easily, is that all the smaller cubes have a red side.

 

At this point one can see that the cube is cut into three slices. The central slice must be as thick as half the original cube's side. It isn't difficult now to answer all questions:

 

1. 32
   
2. 8
   
3. 8
   
4. 28
   
5. 20