Bunuel wrote:

A rectangular carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns. If the height if each can were 4 times greater but the volume was the same, how many cans would the carton fit?

(1) The volume of each can is 60π centimeters square.

(2) k is 200.

There are a few interpretations to this question, in all of them our guiding approach should be Logical:

To calculate the difference in volume of something 'new' and 'old' we need to know the relation between their heights/widths/lengths/radii.

We'll look for statements that give us this information, a Logical approach.

If the carton has a lid, i.e. we care that the cans aren't 'above' the height of the carton, the answer is (E):

If the carton's height is less than 4 times the cans' height, it cannot hold even 1 can! (as the taller can is too tall!)

If the carton's height is more than 4 times the cans' height, at least 1 can will fit.

As 0 and (some positive number) are distinct options, none of the information given is sufficient.

If we don't care about height at all, and all of our rows/columns are 'in a grid' (i.e. one under the other, in

square packing) the answer is (B):

As the volume of each can stays the same, increasing the cans' height by a factor of 4 implies reducing the radius by a factor of 2.

That is, every 'short and fat' can takes up 2r*2r space and every 'tall and thin' can takes r*r space.

This means that we can fit 4 'tall and thin cans' instead of every 1 'short and fat' can.

Meaning that no matter what the actual volume of the cans or of k, the number of cans increases by a factor of 4.

So if k = 200, the new box fits 4*200 = 800 cans. (1) does not give the value of k and is insufficient.

The last option is that we don't know if our rows/columns are 'in a grid' or not. In this case the answer is again (E):

Doing the actual math is a bit involved, but as opposed to square packing where the number of cans is exactly 4k, in hexagonal packing (see the above link) it is a bit less. You can figure this out by drawing a picture and SEEing that you can't fit 4 new cans instead of one old one.

So in this case the new number of cans could be 4k or could be a different number.

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