I need help Solving this
If a 1-inch cube of cheese is cut in half in all three directions as shown above, then the total surface area of the separated smaller cubes is how much greater than the surface area of the original 1-inch cube?
:. there are four smaller cubes in each side of the cheese cube each have a (0.5*0.5) square inches surface area.
(A) 2 square inches
(B) 4 square inches
© 6 square inches
(D) 8 square inches
(E) 12 square inches
the answer is ©
First, a slower, brute force way: count up all the area:
The original cube had 6 faces, each 1x1 = 1 sq inch for a total of 6 sq inches.
Now we have 8 little cubes. Each has 6 faces, .5x.5=.25 sq inches for a total of 1.5 sq inches each. But there are 8 of them…8x1.5=12 sq inches, 6 more than what we started with.
But here’s a better way: each of those little cubes has 6 faces. 3 of them were originally exposed and 3 of them were revealed when the big cube was sliced up. So the new surface area is double the old…
^^ The difference between arithmetic and mathematics. I always did arithmetic. Nice solution & excellent explanation, @pckeller
@pckeller @AboutTheSame how do you get 8 little cubes. There is 1 originally right.
Cut it in half there are 2.
Cut it in half there are 4.
How do you cut it and get 8??
I get how you work it out after finding that there are 8 cubes, but unsure how you figure out it is 8?
You’ve missed one dimension. Put the cube in front of you. Cut in half across the top. Turn it 90 degrees and cut in half again. Yes, you now have four pieces – but they are RECTANGLES, not cubes. You don;t get cubes until you flip the thing on its side and cut in half on the previously uncut surface – which gives you 8 little cubes.
@AboutTheSame I still don’t understand, on the diagram referenced it only shows four pieces no? If there are 8 pieces, then they surely can’t have the same length?
Do you mean like an x y and z axes?
I have been stuck on this question for three hours omg.
Go get some little cubes and try this. Sugar cubes will do it. Try to build a bigger cube, twice as long as one of the smaller cubes. You will soon discover that it takes 8 little cubes!
Think about it this way: the bottom layer will have two rows of two cubes, a total of 4 cubes. But then you will need 4 more cubes to build the top layer.
Then think some more and you will realize that there is a reason they call 2^3 “two cubed”!
And for practice, go ahead and build a cube that is 3x as long as your little sugar cube. You will see that it takes 3x3=9 cubes to build EACH tier. And there are 3 tiers. So takes 27 cubes overall. And 3 “cubed” is 27. Coincidence? I think not…
Look carefully at the diagram. You can see the surfaces of 7 cubes. The 8th one is hidden but you can “see” [infer] that it is there.
Let me try again. This may be a spatial thing. You have a cube of cheese. You cut DOWN through the middle. You cut DOWN again through the middle perpendicular to the first cut. You cut again PARALLEL TO THE FLOOR. Until you do that, the resulting pieces cannot be cubes.
Yes, you have to cut on each axis.
Alternatively (I hope this is not more confusing rather than less):
2^0 = 1 [the uncut cube]
2^1 = 2 [cut in half the first time]
2^2 = 4 [cut in half the second direction]
2^3 = 8 [cut in half the third direction]
Get a piece of cheese and cut it three ways! Then, have a snack. 
@NearMello as @pckeller said, we call x^2 and x^3 “x squared” and “x cubed” for a reason…
And the diagram “appears” to show only 4 cubes, but remember there are 4 cubes behind them.
To add to pckeller’s example: How many cubes would we need to make a 4x4x4 cube? 10x10x10?
@pckeller : brilliant. if you are not a math teacher, you should be.
But I still don’t understand, are you saying that there are 2 rows of four, or four rows of 2.
I completely understand what you are saying after you realise that there are 8 cubes on a face.
The diagram from the question completely bamboozles me.
It appears from the diagram that there are 4 pieces each of unit 1/2 length.
@pckeller May I just thank you for your help btw.
@NearMello : There is a layer of two rows of two, and another layer of two rows of two on top of it.
I’m wondering if we are seeing the same diagram that you are. The diagram I see shows a front face of four cubes, and a three-dimensional representation of four cubes behind it. We only see part of the surfaces of any of those cubes, but use sugar cubes or dice or origami cubes to create the same image and you will see it.
@NearMello The formula for surface area of a cube is 6s squared, where s is the length of the side. (6 because all 6 surfaces are congruent, s squared because that’s how you find area of a square.)
So the original cube has a SA of 6(1 squared) or 6 square inches…*** (stars so you can find this number in a minute.)
Now it’s cut-- each side is split into 4 congruent, smaller cubes. So you have a stack of 8 cubes-- 4 on top and 4 on the bottom, each .5x.5 x5. (picture a Rubic’s Cube, with only 4 small faces on each side.)
The surface area of each of those 8 cubes is 6 (.5) squared or 6(.25) or 1.5
Add up 8 of them, and it’s 12 square inchesl.***
So the question is asking how much 12*** is than 6***… the answer is 6 square inches.
Does that help?
@bjkmom @MITer94 @AboutTheSame @pckeller
Thank you all so much, your explanations were perfect I now understand it.
and @MITer94 for those examples are the answers 64 and 1000?
By George, he’s got it. “Oh frabjous day, callooh, callay, he chortled in his joy.” With apologies to Lewis Carroll.
Ummm, it says on the top of that document that unauthorized copying or using of this document without consent is illegal.
You might want to ask a mod to remove the link.
@bkkmom sorry yes.
Right so the 8 cubes do they represent one of original 1 inch cubes face?
I completely see that there are 8 cubes now. But 1 layer has a surface area of 1. 1*6 = 6 (6 because of the 6 faces).
If there are 8 cubes used (which I know there are), then for one side only 4 cubes appear. So 0.5^2 *4 = 1.
1*6=6.
For anyone else that is reading, by using the answer choices and equating it to the new surface area, you can rule out A B and D, because they do not have an integer for their sides.
You have just correctly calculated that if you re-assemble the little cubes to make one big cube, that cube will still have the same surface area the original big cube started with! But that is not what they asked.
Now that they are 8 separate little cubes, there will be more surface area – sides that were hidden are now exposed to the surface. That’s what they are asking about.
Each little cube: area = 6 x .5 x .5 = 1.5 square inches.
And there are 8 of them so: 1.5 x 8 = 12 square inches.