You are trapped in a circular field of unknown diameter with a low wire fence around the perimeter. Attached to the fence is a vicious and very hungry dog that is free to move along the fence, but not away from the fence. The dog can run four times faster than you can run, and always moves to minimise the distance between you and it.
Can you escape, and if so, how?
Let us first see what happens if you try the obvious - start in the centre of the circle and run in a straight line away from the dog towards the edge of the field.
You will run a distance of R (the radius of the circular field) and the dog will run half the diameter of the field (half of 2 π R) which is about 3.14 R. The dog runs 3.14 times as far as you do, and at four times the speed, so you don't escape that way.
So how far can you run while the dog run halfway around the field? Well, it does not matter how big the circular field is (within reason), so let us assume it is 200 metres in diameter. In this case R is 100.
Half a lap for the dog is 314 metres and it runs four times as fast as you do. Therefore you are able to run 314 / 4 metres while it is running halfway around the field. That is 78.5 metres.
So if you can get to a position 78.5 metres from the edge of the field, or more practically 21.5 metres from the centre of the field, with the dog on the opposite side, you can escape.
The problem with this is that if you start walking away from the dog, it will immediately run around the perimeter to get closer to you. So how do you get you and the dog on opposite sides?
As the dog runs four times as quickly as you do, if you run around a circle 25 metres from the centre the dog will be able to exactly keep pace with you while running around the perimeter. But if you are closer to the centre then the dog will not be able to keep up with you.
So, if you position yourself somewhere between 21.5 metres and 25 metres (say 22 metres) from the centre of the field and then run a circle of that radius, the dog will run around the field after you, but not be able to keep up. Eventually it will be on the opposite side of the field from you, and you take your opportunity to run away from it to the edge of the field.
With the problem as stated, with a field of unknown diameter, you would have to pace out the radius (taking care to keep away from the dog) and then work out 22% of the radius.