There are $50{,}000$ groundhogs on the Isle of Tork. Due to the isolation of the island $1{,}500$ miles from the nearest mainland, it had been spared the scourge of fleas, until a passing sailor on a round-the-world cruise stopped there to walk her dog. On its walk, the dog managed to pass fleas to four groundhogs. Although the dog is gone, the fleas have multiplied and they're spreading to the other groundhogs. In this case, the rate of infestation will be proportional to the product of the number of groundhogs who do have fleas and the number who don't. Let $f$ represent the number of groundhogs who have fleas after $t$ days.

1. Write a differential equation that models the spread of the fleas.

2. Find a general solution to the differential equation.

   Steps (b):

   By partial fraction decomposition:
   After integrating:
   Keep rewriting to solve for $f$.

3. Recall that $f = 4$ when $t = 0$ because four groundhogs were initially infected. Suppose that after ten days, $2000$ groundhogs had fleas. Find a formula for $f$ in terms of $t$.

   Hint (c):

   $4=\frac{50000A}{1+A}\text{ and }2000=\frac{50000Ae^{500000k}}{1+Ae^{500000k}}$