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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?

Ladder and Cube

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?

More or Less?

Are these estimates of physical quantities accurate?

Biology Measurement Challenge

Age 14 to 16 Challenge Level:

The average dimension for each of the following objects is given in the table below:
 

  Length Cross - Sectional Area Volume Modelling by which solid?
Mitochondria 1 $\mu$m 0.79 $\mu$m$^2$ 0.79 $\mu$m$^3$ Cylinder
Arabis voch pollen 30 $\mu$m 706.9 $\mu$m$^2$ 14137 $\mu$m$^3$ Sphere
Ring stage of Plasmodium falciparum 1.5 $\mu$m 0.79 $\mu$m$^2$ 0.03 $\mu$m$^3$ Ring
Tuberculosis bacterium 2 $\mu$m 0.12 $\mu$m$^2$ 0.24 $\mu$m$^3$ Cylinder
Human red blood cell 8 $\mu$m 50 $\mu$m$^2$ 100 $\mu$m$^3$ Disc
Human nerve cell 2 $\mu$m 3 $\mu$m$^2$ 0.3 $\mu$m$^3$ Disc
The eye of a needle 1 mm 1 mm$^2$ 0.1 mm$^3$ Cuboid
Cat hair 3 cm 0.3 mm$^2$ 9 mm$^3$ Cylinder
Snowflake crystal 1 cm 0.79 cm$^2$ 0.17 cm$^3$ Sphere

 

Therefore, we can roughly rank these objects by length, volume and cross - sectional area.