Problem 256: Smart ass
Anyone with a calculator worth its salt should be able to figure this one out.

Despite his many contributions to modern geometry, Thales (tha’les) of Miletus — Greek philosopher born in 624 BC — had to work for a living. One of his jobs was trading in salt.

Legend has it that on an ill-fated journey, one of Thales mules slipped in a stream. When the animal righted itself, it realized its load was lighter, as some of the salt dissolved. Hoping to achieve the same result, the mule rolled over at the next ford and found its load lighter still. By the third of fourth stream, Thales was on to the animal’s tricks. “I’ll fix you,” he said, as he loaded the mule with sponges and rags intended to absorb water and make the load heavier, instead of lighter, after the next dip.

But something unexplainable happened. The mule figured out that if it stayed in the water just long enough, its load-lightening method still worked. How long does Thales mule have to stay submerged to minimize its load, and how much weight is it carrying afterward?

Assume the sponges are weightless, but as they absorb water, they add pounds at a rate of 0.5t1.2. The salt, on the other hand, which starts out as a 50-pound pack, initially absorbs water at a nonlinear rate, gaining 7 pounds in 12 seconds. Once saturated, it begins to dissolve, losing weight exponentially at a rate of e-0.7t, where t is in minutes.

Solution to problem 255, March 2002: Battle hymn bump