Archimedes and Numerical Roots

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?

Interstellar

Weekly Problem 22 - 2012
What number could replace * so that */5 is between 3 and 4?

More or Less?

Are these estimates of physical quantities accurate?

Time to Evolve

Stage: 4 Challenge Level:

We had an interesting anonymous solution to this problem, giving rise to some sensible numbers. We think that similar investigations would make for interesting data-handling project work.

We used the internet to research lifespans. We were surprised to see that fishes and reptiles live for a very long time. A modern day bony fish is the sturgeon, from which caviar is taken. Lizards seem to have quite a wide range of lifespans (between 10 and 50), so we picked 30 as an average. The numbers for dingos and lemurs seemed typical for these sorts of creatures. We also changed the human numbers because in the past both would be smaller. Our table became

@
 Creature Timespan Time to maturity Lifespan Min number Max number Average number Fishes 50,000,000 20 100 500,000 2,500,000 833,333 Amphibians 50,000,000 20 100 500,000 2,500,000 833,333 Reptiles 100,000,000 2 30 3,333,333 50,000,000 6,250,000 Early mammals 125,000,000 2 10 12,500,000 62,500,000 20,833,333 Mammals 60,000,000 2 25 2,400,000 30,000,000 4,444,444 Apes 14,000,000 2 35 400,000 7,000,000 756,757 Humans 1,000,000 13 40 25,000 76,923 37,736

To get the minimum number of ancestors for each row we divided the time in years by the lifespan. To get the largest numbers of ancestors we divded the time in years by the time to maturity.

This gave grand totals of between $19,658,333$ and $154,576,923$ ancestors.

To get the average age of an ancestor when giving birth we found an average age = (Lifespan + time to maturity)/2 and timesed this by the timespan.

Assuming that ancestors were born at an average point in their parents' lives, we get $33,988,937$ ancestors.

Rounding up to sensible values, we think that there were at most $150$ million ancestors, at least $20$ million ancestors and probably about $35$ million ancestors.