Maths Puzzle: Sugar Cubes (orders of magnitude)

Continuing the current series of puzzles taken from Math-E-Magic, here's one which tests understanding of orders of magnitude.

Sugar Cubes
The Big Sugar Corporation wants to persude people to use lumps of sugar, or sugar cubes; so they run a puzzle competition and the first person to achieve correct answers to all three questions wins a lifetime supply of sugar.  It's not the healthiest competition, but here we go.

You have been sent one million cubes of sugar; each cube is just half an inch long, wide and high.

1.  Suppose the 1,000,000 sugar cubes arrive packed into one giant cube.  Where would you store it?  Garage? Warehouse? Under the table?

2.  Suppose you lay the 1,000,000 sugar cubes out on the ground in a single square layer.  How much area would you need?  A tennis court?  A football field?  An entire car park?


3.  Suppose you stacked the sugar cubes (if you were able to play this sweet Jenga) into a single tower 1,000,000 cubes high.  How far would it reach?  As high as a house? Skyscraper? Mountain?  To the Moon?

Let's do some maths.

1.  If we form a cube from the little sugar cubes, then each side of the cube will need to be the cube-root of 1,000,000 cubes long.  The cube root of a million is 100, since 100 cubed = 100 x 100 x 100 = 1000000

Each cube is a quarter of an inch long (we started in imperial measurements, so we'll stay there for now), and since 100 x 0.25 = 25, then the cube will be 25 inches long, wide and high.  Under the kitchen table will do fine.  (25 inches is approx. 60 cm.).

2.  To form a square from the sugar cubes, each side will have a length equal to the square root of 1,000,000 and that's 1,000.  1,000 x 0.25 inches = 250 inches, or nearly 21 feet.  A square with sides of 21 feet would fit into a medium-sized garden, or about six car park spaces.  A tennis court is 27 feet wide (for singles) and 63 feet long, so it would fit comfortably in half of a tennis court (from the baseline to the net).  In fact, the service line (the area which marks the two service boxes) is exactly 21 feet from the net and parallel to it.

3.  One million sugar cubes, eventually stacked one on top of the other, would be 250,000 inches (a quarter inch multiplied by a million).  250,000 inches is 20,800 feet, or 6,944 yards, which is just under four miles.  That won't reach the moon (250,000 miles away), it's more like a small mountain.

Gobally, the top 100 mountains are all above 23,000 feet, but we are still talking around the scale of a mountain.

The highest mountain in the UK is Ben Nevis, which stands at 4,411 feet.
The tallest in Europe is Mount Korab in Albania, which is 9,068 feet.
Sugarloaf Mountain in Brazil is 15,000 feet (396 metres).
Our sugar mountain, at 20,800 feet, is certainly a respectable mountain.

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Incidentally, if you were (in some way) able to stack the sugar cubes at the rate of one a second for a million seconds, that would be 277 hours, or 11.5 days.  If you stacked them at the rate of one a minute (which it may be as you start climbing an adjacent mountain), then you'd be going for almost two years.