Cowculus – the farmer and the cow

The farmer needs to walk to the river, fill up his bucket and then deliver this to the cow. The farmer walks in a straight line from 𝐹 to a point 𝑥 on the line 𝐴𝐵. He then walks in a straight line from 𝑥 to 𝐶.  The problem is to find the best point 𝑥 such that we minimise the total distance walked.

Solving the simple case

We can make an equation f(x) for the total distance travelled:

Let’s use a graph to find the value of 𝑥 that gives a minimum:

This tells us that when x = 1 we get a minimum distance of 8.94.  We notice that this x value is exactly one quarter of the distance from AB – perhaps this provides a general rule?  So next let’s consider a more general case, when the distance AB is m.

Solving a more general case

Now equation f(x) for the total distance travelled is given by:

This time we’ll have to differentiate and set our gradient equal to zero to find any minimums:

Now this requires a bit of algebraic rearranging.  Firstly I move one of the fractions to the other side, then square then rearrange to give:

Next I can factorise and rearrange:

This does then with a bit more work give the positive solution as:

As hypothesised the best place to aim for is exactly one quarter of the distance along AB.

Thinking about this problem is a different way

The nice aspect of this problem is that we can redraw the geometry to create an equivalent problem (which is significantly easier to solve).  The farmer, F, now begins on the alternate side of the river to the cow with AB 𝑚 km and walks in the straight line path FC.  This is shown below:

Clearly the shortest distance in this case is the straight line joining FC.  We can then consider the point where we cross the river as x.  The triangles AFx and xBC are similar with a scale factor of 3, and this leads to the distance Ax being 1/4 of the distance AB – i.e m/4.  We see that the total distance FC will be the same as for the minimum found for the previous problem. So with some creativity we can solve this without any need for calculus!

Changing the initial problem – minmising the time taken

In the next post I’ll look at more significant changes to the problem (where we change the slope of the river) – but for now how about if we change the speed at which the farmer walks when carrying a full bucket compared with an empty bucket.  Clearly this is a more realistic scenario.  Say for example we have the following: