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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?

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?

Time to Evolve

How many generations would link an evolutionist to a very distant ancestor?

More or Less?

Age 14 to 16
Challenge Level


A set of estimates of various physical quantities is shown below. For each estimate, consider the following three questions:

  • Do you think that the estimation will be an over- or under-estimate, or will it be essentially exactly correct? Or will it be impossible to say without more information? Be as clear as you can with your reasoning.
  • Why do you think that the solver made the estimate in this way? What assumptions were made? Can you reproduce the calculation?
  • How close do you think the estimations would be to the real values?
Note that you might need to use standard scientific data not provided in the questions for some of the numbers used and you might need to use a calculator. You will certainly need to decide which formulae to use to relate the quantities in the question.

  1. I wish to estimate the volume of an apple. It weighs 76.2g. I therefore estimate its volume to be 76.2cm$^3$.
  2. I have a set of ball bearings of volume 1cm$^3$. A large crate is filled to the brim with ball bearings and closed with a lid. The box contains 850 ball bearings, so I estimate that the box has a volume of 850cm$^3$.
  3. An oak tree measures 106cm around the base, corresponding to a cross section of 0.0894m$^2$. I have measured the height to be 7.3m. I estimate the volume to be 0.65m$^3$. How would your answer differ if the question related to a fir tree (with the same numbers)?
  4. In a wood of area 27000m$^2$ I find 17 earthworms in a volume of soil of 1m$^2$ in area and 20 cm deep. I therefore estimate that there are 459000 earthworms in the wood.
  5. A cheetah can run at 100km h$^{-1}$, so in a 3 minute chase of prey I estimate that it might be able to run 5km.
  6. A 10 kg sample of sea water taken from a point far from the shore contains 350g of salts. Therefore, to obtain 100kg of salts I would need to take a sample of 3000kg of sea water.
  7. A cylindrical container barrel has a capacity of 700l. To obtain the 100kg of salts required in question 6, I would only need to fill 4 barrels.
  8. A bacterial culture initially contained 7 cells. After 35 minutes it contained 14 cells and after 1 hour 10 minutes it contained 28 cells. I estimate that after a further 12 hours there will be around 43.6 million cells.
  9. I catch 100 adult fish from a lake and mark them all with a tag. One week later I catch 100 adult fish from the lake and find that 12 of them are marked. I estimate that there are 800 adult fish in the lake.
  10. Light of a certain wavelength is shone through a solution of plant cells. 90% of this light is absorbed by the solution. If this solution was diluted 1 part solution to 9 parts water then I estimate that 10% of the same light would be absorbed. A second solution absorbs 100% of this light. I estimate that a 10% dilution would absorb 10% of the light.


Making sensible approximations is a key skill used by anyone involved in the application of mathematics. A good approximation can help to catch errors and reduce the time taken to complete subsequent, more refined calculations. A skilled applied mathematician will always be aware of the level of approximation being applied to a problem, will develop a feel for the range of sensible approximations and will know the mathematics required to justify the approximations.