The Fly and the Spider

In considering the Fly and the Spider problem. One obvious solution (28") is illustrated by the red line. Clearly as we think about the problem a solution such as the one illustrated by the green line seems more economical.

Next we consider three variations of the "green line" approach:

Approach 1:

Let angle AnOtopS = angle BnObottomT = x (where O is the center of the circular surface)

Then it follows that:

and that:

and that the distance walked is given by:

from which we can determine the distance travelled by the spider. Consider the following data generated by a spreadsheet for various values of x:

Graphing this we get the following:

Demonstrating clearly that it is not possible to achieve a route that is shorter than the "red" route by this approach.

Approach 2:

Consider the data generated by a spreadsheet for various values of x:

NOTE: the best solution for this case is in fact the red path in our original diagram. There is no point in considering the graph here as the approach is clearly less "sensible" than approach 1.

Approach 3:

Consider the data generated by a spreadsheet for various values of x:

NOTE: the best solutions for this case are in fact the red path in our original diagram. There is, again, no point in considering the graph here as the approach is clearly less "sensible" than approach 1.

GSP Sketch

It would be attractive to demonstrate these solutions using a GSP sketch - this has proved to be more challenging than first envisaged. A sketch may soon be added here.

Conclusion

We have a very counter-intuitive situation here - that is we expected one of the "green" routes to be the most economical and yet the "red" route is the most economical.

The question that begs asking is why the person who originally posed the question used the dimensions that they did?

Further Questions

Is the outcome a function of the ratio of the diameter to the height of the can?

Is this counter-intuitive situation a function of the distance that the fly and the spider are from the center of the lid and the base respectively?