I found myself longing for my calculus days, when I could understand these pretty complex word problems without a second thought. Working through these following problems, I struggled with the algebra, setting up the problem, and trying my best to remember how calculus makes these problems so much easier.
The rate of change is a foundation of calculus and derivatives. Since I've forgotten most of that and don't particularly want to reteach myself in a few hours, we'll skip over as much of the calculus as possible. Welsh corgis are excellent at doing calculus, maybe they can review calculus with me.
If you'll remember those math problems that state: a train leaves from City A at some time going at x miles per hour and another train leaving from City B leaves at a later time going at a different speed, the word problems tend to ask "When do they meet?" or "Which train gets there first?" These are the problems I'm talking about, distance = rate x time or d=rt. While they're fun to work through, they take up a lot of time and mental concentration to get all the parts right.
If train A and train B are traveling on the same track, shouldn't the question be "When do they crash?"
I worked through this problem using paper and pencil and found that I struggled with the alegbra. Despite my rusty algebra and calculus skills, I found working through the problem to be relaxing. Plus I couldn't stop once I started. The question asks if it's faster to run half a distance then walk the other half, or to run half the time and walk half the time.
This problem was also interesting to read through. I tried to maximize her sponsorship money, but after graphing the equation I realized that she would make the most money if she ran the entire way, but that's not realistic, or what the question was asking. Oh well.
Back to my initial interest in this topic. As a walker in Boston, we've all experienced getting stuck behind someone who is taking their time looking at buildings and the scenery. However, when you're in a rush or trying to catch a bus or T, I severely dislike it when a slowpoke is in front of me on the sidewalk or walking down the stairs. We all walk at different speeds, and I shouldn't blame anyone but myself for being late, but after doing some research, it seems that rates of walking are much more mathematically complex than it would appear.
Effectively navigating the streets of Boston takes a lot of skill.
Walking sometimes requires an optimization analysis, especially if you're running late. Let's say I've got somewhere to be at 6 but leave the office a later than I planned. Should I walk to the nearest T or should I just walk it to the station I would be transferring to later? It ultimately depends on how fast you walk versus how long a wait would be for the T. Other factors like weather also influence people's choices. People cut through grass fields, take smaller roads, use alleys for shortcuts, use the stairs instead of wait for the elevator, etc. Path optimization is anoter part of math on foot.
I also think about judging speeds as a walker. If you're crossing the road when the crosswalk lights haven't said it's ok (jaywalking if you will) you have to properly judge how fast a car is going. You might also have to judge the rate of change in speed if the driver chooses to slow down and let you pass or speed up to try and hit illegal street crossers. I'll try my best to stop jaywalking, but in the meantime, I'll use my rate judgement to stay alive. This goes to show survival skills depend on math.
So today's blog post was brought to you by: rates of change, distance, rate, time, and speed.
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