Doubt I Greg QQ bank

GRE Quant Problem Solving
1

Hi! Did not understand the logic of this explanation:

image
image1005×463 67.5 KB

Here, Greg says that to make the cylinder maximum, the rectangle must actually be a square.

Now what I did was, take any rectangle box (as that is what is given in the QQ), and I maximised the cylinder value possible- and then just found unused area.

Can you please explain the logic of taking a square in some more detail please?

2

image
image942×896 15.9 KB

Let us consider any arbitrary cyindrical container and is tightly packed in a rectangle box as in the above image. Considering the radius of cylinder as r and heigh as h, will give us the box height as h with both length and width being 2r.

Cylinder volume = pi.r.r*.*h = pi.h.r^2
Box volume = (2r).(2r).h = 4.h.r^2

Unoccupied space percentage = 100 * (Box - Cylinder)/Box
= 100 * (4 - pi)/4 = 21.5 (approx)

Alternative:

It can be noted that the dimensions do not matter. So I guess we can take any arbitrary dimensions and end up with the same percentage for the answer.

I hope the calculation and answer are correct. I apologize for the poor diagram and formatting.

3

Thank you for the explanation, this is clear to me.

However, this same calculation is not working with any arbitrary rectangle. Eg., I took rectangle dimensions to be 8X6X2- and in this case, the largest cylinder which is possible to fit will correspond to the 8X6 face, having a radius of 3, with height=2.

The ans. is that case is around 40%- therefore wanted to confirm how I can arrive at the conclusion of strictly taking a square face, and not any rectangle dimensions

4

Oh sorry. I might have misunderstood your query earlier.

You want to understand why we must take a square face. Mathematically deducing that shouldn’t be too complicated.

  1. We are already clear that the height of the box must be same as that of the cylinder. And now we are left with the other two dimensions.
  2. This can be reduced now to imagining in just two dimensions - and the problem is now just fitting a circle (the base of the cylinder) in a rectangle (the base of the box)
  3. The smallest rectangle we can cover the circle is with a square with side length same as the circle’s diameter touching it from all sides. Any other rectangles are always larger than this square and will give more space for circle to move in the rectangle.

#: An example from real life, if it helps, can be that cylindrical water bottles are packaged/delivered in rectangle boxes and they are square-base ones. If it were any other rectangle or larger square that would allow the bottle to tumble in the box.

Hope this helps!

5

Got it, thank you so much for this detailed explanation