Today I thought I'd do a little mathematical analysis of Hawaiian Air's luggage handling fee scheme using some simple calculus. Hey, wait, keep reading! I promise I'll try to keep it interesting, and the formulae to a minimum.
I don't remember when I first started wondering about the luggage fee scale used by Hawaiian Air, but it was probably the first time I was checking my own bags in and discovered that the fees go up – way up – upon checking more than one or two bags. But they didn't vary in any sort of predictable way that I could see, so I took the liberty of plotting them out, as you can see below:
Now, while this tells us how fast the fee is changing per bag, we can get more information out of it by taking a second derivative and find out how fast the rate of change is itself changing. For example, it looks on the graph like the biggest jump in terms of absolute price is between the second and third bag, and if we take the second derivative (essentially taking the slope of the points formed by taking the first derivative) we see that this is indeed the case. Taking the second derivative, we see that the price increases by $10 dollars for the second bag ($35 instead of $25), $90 for the third bag, 0 for the fourth, fifth and sixth ones, $75 for the seventh, and 0 for the eighth again. Minima and maxima (the largest and smallest values) of the second derivative tell us exactly where the function is changing its value most rapidly, so we can see that the biggest change is between two and three bags.
So what's the lesson we get from all this? First, calculus isn't very difficult, and knowing it can enrich your life in many unexpected ways. In fact, calculus is like a fruit tree; you spend time planting it, watering it, and nurturing it, and in return it continues to delight you with delicious fruit long into the future. In many ways I think calculus is simpler than algebra, trigonometry, or geometry, yet so many people are afraid to try it because of preconceived notions about their mathematical ability. To put it another way, calculus is like color vision; neither is essential to life, yet lack of either leaves the world a drab and dreary place.
Secondly, try to avoid taking three bags on Hawaiian Air if you can, it just gives you the worst bang for your buck, so to speak. Stick with two, or one if possible.