An annoying little problem...

"A farmer plouged a square field, ABCD, of side 132 metres.

There is a path along the perimeter of the field which the farmer can walk along at a speed of 8km/h. He can walk across the plouged field at a speed of 5km/h.

In order to get from A to the opposite corner C, the farmer starts walking along the perimeter path from A to B. When he reaches a point P he leaves the path and heads directly for C, across the ploughed field.

What is the distance. AP, if he takes the least possible time in getting from A to C by the route described."

Now I got a solution for this what I done was let BP=x so that the distance AP would become 132-x and PC would become

\sqrt{x^2+132^2}

. I then made it a time function by putting

\frac{132-x}{8} + \frac{\sqrt{x^2 + 132^2}}{5}

. Differentiated it and whatnot, solved it for x at the minimum value and ended up with x=105.68

However, I'm pretty sure that I'm wrong with this but I'm not exactly sure where. The problem seems to lie in that fact that the function I have differentaited only describes when x>0, it doesn't take into account the farmer just walking right round the perimiter. The worrying thing is that it seems to be a fair bit quicker for the farmer just to do this, which leads me to believe that my function for the time is wrong. And I was expecting to have to use the product rule/quotient rule at some point, because this is what we've been studing lately (optimisation is part of the course, but this question could have been given to us last year but apparently it's difficult for this year, which also leads me to believe I'm doing something wrong.)

Any help is much appreciated.

Reply 1

steve2005

Using Excel I think AP is about 26.32 metres.

Journey time for AP = x/8

Distance across ploughed field =(( 132- x)^2 + 132^2)^0.5
Journey time across field = ((( 132- x)^2 + 132^2)^0.5)/5

Add these two times and plot against AP. U shaped curves produced with minimum time occuring when AP = 26.32 metres.

I hope the above is OK.

See attachment.

ploughed field.xls48.1KB

Reply 2

steve2005

Edit

I have just noticed that your 105.68 plus my 26.32 = 132

Reply 3

illvolution

Thanks for helping man. I would have no idea how to solve these kind of problems using a computer, but that seems like a nice way to do things.

But yeah, it seems that your results are in agreement. When I done the question I decided to try and solve for BP and take this away from 132, so we've both found the same thing. I was just looking at the question again just now, it seems I've been a bit of an idiot, it seems to quite explicitly say that the farmer has to go across the field, I think I may not have to worry about the farmer walking round the perimiter at all.