The time it takes for either to get to the desired edge of the pool is proportional to distance they must travel (time=distance/speed). The criminal must travel distance r (radius) to get to the edge of the pool farthest from the police officer. Naturally, the criminal will swim to the edge farthest away from the police officer. If the police officer were to run all the way around the pool it would be a distance of 2*pi*r. In this case, however, the police officer will only need to run half the circumference of the pool, which is 2*pi*r/2 or pi*radius.
The ratio of the officer’s distance to the criminal’s distance is pi*r/r which simplifies to pi/1 which is about equal to 3.14/1.
The ratio of the officer’s speed to the criminal’s speed is 4/1 (this value is given to us).
Using this ratio and the equation time=distance/speed we can set up a ratio for the officer’s time get to the opposite side to the criminal’s time to get to the edge of the pool:
(pi/1)/(4/1)=pi/4 which is about equal to 0.785
We can clearly see that 0.785 is less than 1! This means that the ratio of the police officer’s time to get to the desired edge of the pool is less than the criminal’s time to get to the edge. So, the answer is yes, the officer does catch the criminal.
Although, if you're being picky about it, the answer would be no, the officer cannot capture the criminal BEFORE he escapes from the pool because the officer can't swim. He can, however, capture him AS he is escapting from the pool, haha.
