I don't own many pants, and the pants I do own get a lot of wear and tear, through their constant contact with my oversized backside, and the endless washing such contact necessitates. When I first bought these khakis, they had a mass of 450 grams. After 45 days of constant use, however, the mass had decreased to 415 grams. Assume that the rate of decay of the pants' mass is proportional to the mass of the pants, and that the mass is decaying from all parts of the pants equally, not just from the butt area, which is far more likely. I have determined that if the pants' mass drops below 190 grams, they will no longer block out my naughty bits, and they will begin their long, slow decline towards transparency (which benefits no one, trust me). How many days from the date of purchase do I have until I cross the danger threshold of 190 grams?

Difficulty:

Solution:

The key to solving this problem lies in the fact that "the rate of decay of the pants' mass is proportional to the mass of the pants." If something's rate of change is proportional to the something, itself, then you can assume exponential growth (or, as is the case here, decay) is present. This means you use the ironic formula:

(It's ironic because you could pronounce it "naked," but that is just an aside.) In this formula, N represents the original quantity (450 grams of pants), and y represents the amount of pants left (415 grams) after t = 45 days. Out goal is to find k and use it to answer the question posed in the problem. So, plug in your known values and solve for k:

Now we can find the t which corresponds to the final day of the pants' viability (i.e. when the mass is 190 grams). Any day past that point represents danger:

I can count on about 479 days of safe pants-wearing. However, at some point on the 480th day, there's gonna be excitement.