Scarfin' Willie, also known affectionately as "Hoover," always wins the annual hot dog eating contest. But even Willie slows down as he scarfs! In fact, his rate of hot dog consumption is inversely proportional to the number of hot dogs he has eaten.

Ten minutes after the start of the contest, Willie is eating $2$ hot dogs per minute. After 15 total minutes of eating are allowed, time is called and Willie is once again declared the winner. How many hot dogs has Willie eaten?