A 4 x 4 x 4 wooden cube has been assembled using unit cubes. If it is cut at 45 degrees angle starting at one of its facial diagonal, how many of the cubes are cut in the process?

9514 1404 393

  18 cubes

Step-by-step explanation:

We assume that the 45° angle is measured between the cut plane and the face of the 4×4×4 cube. For the purpose of figuring this out we have defined the cube to lie with its upper front corner at (x, y, z) = (0, 0, 0), and its diagonally opposite corner to lie at (x, y, z) = (4, 4, -4). The cut plane has equation ...

  x/4 +y/4 -z/√8 -1 = 0

For a given small cube corner point (x, y, z), the diagonally opposite corner point will be (x +1, y+1, z-1). We have assumed that a small cube is cut if its diagonally opposite corner points lie on opposite sides of the cut plane. The side of the cut plane the corner lies on will be indicated by the sign of the sum ...

  x/4 +y/4 -z/√8 -1

For the 64 corners of interest, we can use a spreadsheet to calculate whether the cube corner and its diagonal opposite lie on opposite sides of the cut plane.

We find that 9 cubes on the top layer are cut, 6 cubes on the second layer, and 3 cubes on the 3rd layer are cut, for a total of 18 cut cubes. The entire bottom layer is untouched by the cut plane, which exits the z-axis at -√8 ≈ -2.8, above the top of the bottom layer at z = -3.

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The attachment shows the spreadsheet output for the top layer of cubes. Even values ±4 indicate the entire cube is on one side of the cut plane. Even values ±2 indicate the cube straddles the cut plane (is cut).

Odd values (±1 or ±3) or 0 indicate a corner of the cube lies on the cut plane. This is only the case for the top plane of cubes, where the face diagonal is the line of intersection.

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Alternate solution

The second attachment shows the 4×4 array of cubes with the cut lines at each layer. There is no cut line between layers 3 and 4, because the cut plane exits the cube above that layer. The cut cubes on each layer are ones that lie between the cut line on the plane above and the cut line on the plane below. (Here, we're using "cut line" to mean the line of intersection between the cut plane and the plane bounding a cube layer.)

is there a picture

step-by-step explanation:

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: