I found a new apartment that I’m in love with, and I’m moving at the beginning of May. I called two moving companies which had been recommended to me to get price estimates.
Both will send 3 people.
Company A charges $100/hr for their services. However, they have a 2.5 hour minimum charge. They also charge $100 for 1 hour of travel time (30 minutes to the apartment and 30 minutes from the apartment). There is an additional fee of $50 for packing charges for my futon, mattress, and TV. Lastly, they charge by the quarter hour. So if they worked 151 minutes, they’d charge me for 2.75 hours.
Company B charges $130/hr for their services. They have a 2 hour minimum charge. They also charge a travel fee, but it is only $65 in total. Lastly, they too charge by the quarter hour. So if they worked 121 minutes, they’d charge me for 2.25 hours.
The question I was left with is: which is the better company for me to hire?
The only variable is the amount of time the movers spend moving.
The x-axis is the hours spent moving, the y-axis is the amount of money my bank account will be missing. Company A is in blue, Company B is in red.
What’s awesome is that we just taught functions and function transformations in my Algebra II class. And one of the functions we worked with a lot was the floor (step) function. We’ve also talked about piecewise functions. If I were teaching an accelerated class, I would literally give them the information and ask them to (a) first produce the graph, and then (b) from the graph, come up with a function that gives them the graph.
They’re going to have to recognize they’ll need a piecewise equation, and then also have to figure out how to make the function transformations on the step function to get the second half of the piecewise equation.
I kind of love this problem.
Continuous Everywhere but Differentiable Nowhere 
Fantastic! Real life examples of step functions are hard to come by.
@Kate: Thanks! If you live in a city, taxis are piecewise in the same way (you pay by the 1/5 of a mile, or something, plus some initial fee like $2.50). I’ll try to think if I can come up with something that isn’t a price…
Air conditioner logic/temperature. It blows at a certain level when it hits 78, then harder when it hits 80, etc. At least, that’s how I think it works?
Taxes are sort of like this (although more complicated). You could come up with a semi-artificial flat-tax that was tiered. Still money, I guess.
Power from your car is also tiered like that…when you change gears, it will jump. (There are TONS of charts on this called “dynos” that measure torque and horsepower based on RPMs for different gears.)