Consider eight two-cubic centimeter (2 cm3) sugar cubes stacked so that they form a single 2 x 2 x 2 cube. How does the surface area of the single, large cube compare to the total surface area of the individual eight cubes? Report your answer as a ratio. Be sure to show all calculations leading to an answer

- anonymous

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- anonymous

@Preetha hey can you help me please

- anonymous

@perl hey can you please help

- mathstudent55

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## More answers

- mathstudent55

Since your answer is a ratio, you don't need to know the specific surface area of each cube. You just need to know how they compare.

- mathstudent55

Look at the small cube to the right.
As any cube, it has six congruent square faces.
Let's call the area of each face x.
What is the total surface area of the small cube?

- anonymous

12cm^3

- mathstudent55

No.
2 cm^3 is the volume of the small cube.
We are not interested in that.
Above, I told to let x be the area of 1 face of the small cube.
The cube has 6 faces, all congruent squares.
If 1 of those faces has an area of x, what is the area of the 6 faces combined?

- anonymous

ohh wait sorry my bad that was me completley overlooking the problem haha

- mathstudent55

Once again, if one face of a cube has area x, what is the area of all 6 sides added together?

- anonymous

6x

- mathstudent55

Great.
Now we need to look at the large cube on the left in the figure above.

- mathstudent55

Since the large cube is made up of 8 small cubes, each face of the large cube is made up of 4 faces of the small cube, right?

- anonymous

i should probably mention this is physics in case that matters

- anonymous

and yes that is correct

- mathstudent55

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- anonymous

24x

- mathstudent55

Ok. See the new figure above.
One face of the small cube has area x.
One face of the alrge cube has area?

- mathstudent55

^large

- mathstudent55

I see. You're ahead of me.
Correct.
Each face of the large cube has area 4x, and the entire cube has a total surface area of 24x.

- mathstudent55

We now have:
Surface area of the small cube: 6x
Surface area of the large cube: 24x
Now express the ratio of the area of large cube to the area of the small cube using 24x and 6x.

- mathstudent55

Fill in the second fraction:
|dw:1433805590790:dw|

- anonymous

6:24, 1:4

- mathstudent55

That is small to large.
I would do large to small.
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The ratio of the large cube's area tot eh small cube's area is 4:1

- mathstudent55

^to the

- anonymous

ok so is that all that needs be done

- mathstudent55

Yes.
The business about 2 cm^3 volume is too much info to throw you off.

- anonymous

ok lol well thank you

- mathstudent55

Now I would just like to make an observation.
We worked with areas.
Now let's look at volumes.

- mathstudent55

The small cube has a volume of 2 cm^3. The large cube is made up of 8 small cubes, so its volume is 8 * 2 cm^3 = 16 cm^3, right?

- mathstudent55

Also, notice that the side of the large cube is twice the side of the small cube.

- anonymous

i have a question

- mathstudent55

When the side of the cube was doubled, the area became 4 times larger, and the volume became 8 times larger.

- anonymous

no i understand that what i don't it how you can tell whether 2cm was the surface area or the volume

- mathstudent55

Notice that 4 is 2^2, and 8 is 2^3.
This is a general rule helps to know:
If a solid changes in size by a scale factor of k, the area changes by k^2, and the volume changes by k^3.

- mathstudent55

A centimeter is a unit of length.
Think of a centimeter as being used the same as an inch.

- mathstudent55

A centimeter is smaller than an inch, but both a centimeter and an inch are units of length.

- anonymous

excatly and volum is a measure of the area of the entire space a object takes up which i didnt think could be measured by length

- mathstudent55

For example, a regular piece of paper measures 8.5 inches by 11 inches.
The length and width are measured in inches.
That means the length and width can also be measured in cm.

- mathstudent55

An area is the size of a surface.
If you measure a length and width in inches, then the unit of are is the square inch, or sq in., or in.^2.

- mathstudent55

If you measure length and width in cm, then the area is in cm^2, or square centimeters.

- anonymous

like yes volume is length time width times hight but seeing as how we only have 2cm^3 shouldnt that be the surface area of a side

- mathstudent55

This is where this problem gets tricky.
We were never told the side of the cube.
We were only told the volume of the small cube is 2 cm^3.
Since length, width, and height are all measured in cm (or inches), when you multiply the length, the width, and the height of a cube together to find the volume, you are multiplying three lengths in cm. You get the units of cm^3.

- anonymous

i have to go eat can we pick this up in 30 minutes

- mathstudent55

The fact they told us the cube was 2 cm^3 means that must be a volume.
The units of cm^3 are a giveaway that this measurement of 2 cm^3 must be a volume.

- mathstudent55

The reason I call the problem tricky is that the volume was never needed.
Since we were only looking for a ratio of areas, the volume was unnecessary.
On top of that, they gave us a volume that you cannot easily show the side as a nice even number.
Since a cube has a volume of 2 cm^3, that means the side of the cube is the cubic root of 2 (with cm as the units).
\(\large V = 2 ~cm^3\)
\(\large side = \sqrt[3]{2~cm^3} = \sqrt[3]{2} ~cm\)
The cubic root of 2 is an irrational number that if written as a decimal is a non-terminating, non-repeating decimal.
This is why I think the problem is tricky. t gives unnecessary info that is hard to write as a number.

- mathstudent55

I also have to go.
If you have questions, ask, and I'll try to answer next time I'm on.

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