Time for a child to ask "Are we there yet?"

I was asked by Rebecca Wallace representing Skoda Auto if I could formulate an equation for the time for a child to ask The Question

``Are we there yet?''

She was interested in the dependence on the number of activities and the number of children in the car, as well as the length of the journey. I agreed to help as a fun exercise for people to think about as they set off on holiday with their children.

My best guess for the simplest reasonable equation involving these quantities is

$\fbox{\parbox{2.5in}{\begin{equation}T = t_0 + \frac{1 + \beta A}{\alpha C^2} \nonumber\end{equation}}}$

where

Listen to
The World Tonight
interview
from 21 July, 2006
T is the time at which a child first asks ``Are we there yet?''
is the time we left (the house)
C is the number of children in the car
A is the number of activities they brought to do
and are parameters. (See here for further information)

Why this form?

The equation contains A. This means that the length of time the children are quiet grows with the number of activities in the car. The reason the equation contains 1+A rather than simply A is that even with no activities, A=0, children will not ask The Question immediately - one can generally at least get out of the driveway.
The more complicated part of the equation to work out and to explain is the dependence on the number of children. The equation involves because this is a simple approximation to the number of interaction between C children.^* This is well known to parents - the number of arguments between children is not proportional to the number of children. The change from 2 to 3 children more than doubles the number of possible arguments. By putting in the denominator of the formula I am assuming that, at least after a certain amount of time, children primarily argue and this increases the likelihood of someone asking The Question. It is of course possible that the should go in the numerator. This would correspond to children happily playing together in the car. A quick poll taken in the Warwick Mathematics Common Room suggested that the denominator was more likely to be correct.

Note that the equation contains the fact that if there are no children in the car (C=0) then the time goes to infinity. This is correct since if there are no children in the car then one assumes that nobody will ask ``Are we there yet?''.

Disclaimer

This is meant to be fun and to help people think about mathematics. I do not expect the equation to be able to make accurate predictions for your particular children.

^*The actual number of interactions between C children is C (C-1)/2 .

Dwight Barkley 2006-07-25