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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?


Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

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?

Approximately Certain

Age 14 to 18
Challenge Level

Why do this problem?

This problem gives an excellent workout in estimation and calculation using a wide range of physical equations and situations. It is rather open, and will particularly benefit those students used to following recipes in their work. It also highlights the fact that in science it is rather hard to calculate anything without making some sort of assumptions. Good science will clearly state and be aware of these assumptions; bad science will ignore them.

Possible approach

This problem could be approached in two stages. First of all the problem could be discussed as a group without any calculations being made (except in students' heads or on the back of an envelope). Once the issues are uncovered, students might wish to begin calculation or might need to turn to the internet or other resources for more information.

It is likely that students' answers and approximations will vary. Once the task is finished, groups could feed back to the rest of the class. Is their reasoning and explanation clear? They will need to convince the rest of the class that their ordering is correct. This could be done with each group attempting each of the four different sets of data. Alternatively, different groups could attempt different parts of the task, in which case the explanations might need to take on a more detailed focus.

There are two different levels at which the problem might be approached

Basic: Produce a means of calculating each part with various estimated values of the data. Order the answers.
Advanced: Produce definite upper and lower bounds for the quantities, using upper and lower bounds for the input data. The ordering is only guaranteed when these intervals do not overlap.

Don't lose sight of the fact that the ordering is important. If a very crude approimation shows that one of the quantities is clearly largest or smallest, then that is sufficient. Of course, students might be interested in computing a more accurate answer out of general interest.
This worksheet has all the quantities.

Key questions

  • What is precisely stated and what is not precisely stated?
  • What factors would complicate the most accurate calculation? How can we deal with these? Which factors can we neglect and which are important?
  • Can you give quick, sensible lower and upper bounds on the quantities before attempting a computation?
  • Is a detailed computation necessary for all of the parts of the problem?

Possible extension

The most able students should be required to approach the task with the most rigour. They might also consider the best way to represent their results. What accuracies are most relevant? Is a linear measurement scale suitable?

Possible support

Focus on the basic method of approaching the task, as mentioned in the possible approach.