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# Approximately Certain

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### Ladder and Cube

### Archimedes and Numerical Roots

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Age 14 to 18

Challenge Level

In this problem we look at several sets of physical quantities. Some of the quantities are very precisely stated and will be possible to work out exactly. Others might be clearly stated, yet will defy an exact calculation. Others might not be clearly stated: you will need to state some more assumptions or do some research before a meaningful approximation might be made (be scientific about
this process).Whilst it might not be possible to calculate exactly each value, each does have a value: **you are required to arrange each of these in order of magnitude** . Be sure to justify your ordering with scientific and mathematical rigour.

The energy:

- Used to walk up the steps of the Burk Dubai skyscraper ($818$m)
- Contained in a full-sugar can of coke
- Contained in a single atom of lead (according to Einstein's equation $E=mc^2$)
- Needed to boil a kettle of tap water

The time taken:

- For a radio wave to travel halfway around the world
- For a top sprinter to run $1$mm at top speed
- For the end of the second hand on a watch to move a distance of $1$ micron
- For a test tube of hydrogen gas to fully combust when exposed to a flame

The distance:

- You could jump vertically up on the surface of the moon
- You could throw a tennis ball-sized lump of lead
- Between peaks of two sound waves caused by two successive hand claps in your fastest possble round of applause
- You can run in $1$ second

The mass:

- Of the atmosphere
- Of all of the people in the world
- Of the north polar ice cap
- Of all of the living bacteria presently living on Earth

Can you make up other similar quantities? Be inventive!

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?