there is a 3 inch cube thats painted red. The length,width,and height is 3 inches long, Answer these questions that come from the info . a.How many cuts will it

Question

There is a 3 in. cube . Its width,length,and height is 3 inches long. answer these questing coming from the information........

• A: How many cuts will it require to divide the cube into 1 inch cubes ?
• B: How many cubes would you have ?
• C: How many cubes would have 4 red sides ?
• D: How many cubes would have three red sides ?
• E: How many cubes would have one red side ?
• F: How many cubes would have no red sides ?

 As you answer parts c and f of the problem, try to think of a way of explaining where the cubes in your answer would be. You might devise a numbering system or another method so that the reader will understand which cubes you are referring to .

Answer 1

Think of this as a visualization problem.

 

 

A)

 

If we cut the width equally 3 times, cut the length equally 3 times, and cut the height equally 3 times, we will have cubes that are each 1 inch.  Therefore, we have a total of 9 cuts.

 

 

B)

 

To find out how many cubes we will have, we multiply 3 by 3 by 3.   We will have 27 cubes.  This is because if you build the 3 inch cube using 1 inch cubes, at height 1 inch the number of cubes is 9.  Repeat this process two more times to get the height of 3 inches.  At height 2 inches, the number of cubes is 18.  At height 3 inches, the number of cubes is 27.

 

 

C)

 

The bigger version of the cube is painted red, so only the faces that were part of that cube will be red.  Those that are NOT painted red are the interior sides of the cube when we cut them up.  The 3 inch cube will have 9 painted squares at each face.  Some cubes exhibit more than one exterior face. For example, the edges of the 3 inch cube.  Therefore, there no cubes that have 4 red sides. This is because at least one side of the cube is an interior side.

 

 

D)

 

Use the method from part C.  Only the corner cubes will have 3 red sides.  Therefore, there are only 4 cubes with 3 red sides.

 

 

E)

 

If we look at the faces of the 3 inch cube, each side will have 9 squares.  However, some squares has more than one red side.  That means the middle square will have one red side.  Multiply this number by 6. 

6 cubes in total have only one side.

 

 

F) 

 

The inner interior cubes will have no red sides.   There are 6 cubes in this position, so 6 cubes have no red sides.

 

Answer 2

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a) To make the cubes, you make two cuts down from the top to the bottom, two cuts from the front to the back, and two cuts from the left/right to the right/left.

2 + 2 + 2 = 6

 

b) We have 3 by 3 by 3: three cubes high by 3 cubes wide, and they're 3 cubes deep:

3 * 3 * 3 --> 3 * 9 = 27

 

c) The most painted sides would be at the corners, the most available place to get paint. If we examine the top right front corner cube, it would have paint on the front side, on the top, and on the right face--three max. The other three sides are covered by the other cubes, so 0 cubes have four red faces.

 

d) Three-red faces would be the corners. There are 8 corners, so 8 cubes.

 

e) One red side would have to be the direct middle place on each side. If we go up/down or to the sides, we hit an edge and thus have two painted sides. A cube has 6 faces, so there are 6 cubes with one painted face.

 

f) No painted sides means we have to go inside the cube. Lucky for us, since the large cube is 3 cubes deep, there is only on in the center (3 - 2 = 1. Now matter from which face, it is 3 deep, and 2 of them are outside and thus red).

This mean there is only 1 cube unpainted.