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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Differential Equation Matcher

Why do this problem

Thinking about building equations will give a real boost to students' understanding of differential equations as dynamical processes in which some quantity undergoes a continuous change.

Students will need to appeal to basic physical principles and the concepts of rates of change to match all of the equations.

Possible Approach

This problem would work well as a card-sorting activity and a starter.

In each equation there are approximations made, which should be discussed as a group.

This could lead to stimulating discussion in which students become aware that the equations studied at school are in fact simple approximations to real world problems. The skill of the mathematical physicist is in understanding when an equation gives good answers and when an equation breaks down and needs refinement.

Key Questions

• What do the equations say in words?
• Do you expect the derivatives to be positive or negative for positive and negative values of $x$? Why?
• Is the assumption that $X(t)$ is any number sensible? If not would it still give reasonable solutions to the modelling problem?
• Are there other effects in the evolution of $X$ which could be added to the equation? Would these effects be small or large? How could these effects be incorporated?
• Is it reasonable that the constant terms are indeed constant, or is it more realistic that they might vary at some point in the evolution of $X(t)$?

Possible Extension

Can students invent their own pairs of process and sensible matching equation?

Possible Support

Try the introductory problem It's Only a Minus Sign