Bunuel wrote:

There is a rectangular prism made of 1 in cubes that has been covered in tin foil. There are exactly 128 cubes that are not touching any tin foil on any of their sides. If the width of the figure created by these 128 cubes is twice the length and twice the height, what is the measure in inches of the width of the foil covered prism?

A. 4

B. 6

C. 8

D. 9

E. 10

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION1) Don’t give up before you start!I can feel your skin crawling as fear sets in, but believe in yourself! Let’s start working like we do with any other problem. This problem is at the end for a reason. You should have finished everything else already so all you need to focus on is this problem. Don’t worry about the time at this point. You will waste more time worrying than you have to spare. Start working! And start with what you know!

2) Start with what you knowWe know: 1 in cubes is the same as inches cubed (volume in inches cubed is essentially a measure of how many cubes that are 1in x 1in 1in that can fit in a 3D shape). We also know that the 128 cubes are completely covered by other cubes so that none of their sides touch the outside. This means that there is essentially a bigger prism completely covering a smaller prism. Its like when you used to play with blocks (I know you did!) and you would completely enclose a block in other blocks to make an exact replica only bigger. So we have a rectangular prism that is 128 inches cubed. Finally, we also know that one side is twice as big as the other two.

3) Draw a picture (if possible)Let’s try to draw what this might look like. If we had a cube that was 2 x 2 inches cubed, one side would look like this:

Attachment:

Ex1.jpg [ 1.44 KiB | Viewed 1639 times ]
In order to cover it completely on all sides we would have to have a cube that had one more cube on each side. So the new face would look like this

Attachment:

Ex2.jpg [ 3.04 KiB | Viewed 1642 times ]
So we essentially know that the final figure will have sides that are two greater than the sides of the interior structure, (again, this would enclose a smaller rectangular prism). Huzzah! We are getting somewhere.

4) Use general equations to get specific answers

Using the equation of a rectangular prism and the information that our smaller prism has a length that is two times the width and height we can start writing equations:

L x W x H = Area of a rectangular prism

Let’s make L = x That would mean that H = x also and W = 2x and we can plug that into our equation to get:

(x)(2x) (x) = 160 or (2x^3) = 128 —-> x = 4

5) ANSWER THE QUESTIONWe are SO close to the end, but we need to answer the question being asked. Now the length and height of the smaller prism is 4 in (which is an answer choice) but that is NOT what the question asks. The width of our smaller prism is twice the length or 8 in. This is also NOT what the question asks! We know from our previous calculations that all the measurements of the final figure are two greater than the smaller so the dimensions of the larger prism would have to be L= (x+2), H= (x+2) and W = (2x+2). Thus our final answer is 2(4) + 2 = 10. WOOHOOO we DID IT!!

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