The Related Rates of an Automatic Pizza Saucer
Posted by Bowman Dickson
I always struggle a bit with teaching related rates, because the focus of all textbook/AP problems is to calculate how fast something is changing in one single moment in time, which short shrifts the beauty of the topic. The main idea seems more to be thinking about how two rates are, well, related to each other. What must happen to one rate as time goes on if another rate stays constant? That’s what makes that classic ladder problem even remotely interesting (why the hell would one side be moving at a constant rate while the other is speeding up?).
I saw this gif of an automatic pizza saucer a while ago and immediately thought it would be a fabulous discussion piece for this very idea:
Someone who designed this used some sort of calculus, even if it was the intuitive kind! We talked about this in class for 15 minutes or so, and the students that enjoy wrestling with not-so-obvious and applied situations really enjoyed thinking about these types of questions:
- If the pizza spins at a constant rate and the sauce comes out at a constant rate, what has to happen to the speed of the arm?
- If the pizza spins at a constant rate and the arm moves at a constant rate, what has to happen to the rate at which the sauce comes out?
- If the arm of the pizza moves at a constant rate and the sauce comes out at a constant rate what has to happen to the rate at which the pizza spins?
We came to some interesting conclusions about the above, including that one of the situations above is not possible (I think!), which we only figured out because one student and I were vehemently defending two different and opposite things.
Anyone want to try to throw some numbers on this to figure out these very questions?
Posted on April 13, 2015, in Calculus and tagged Related Rates. Bookmark the permalink. 3 Comments.
The rate of spinning of the pizza is irrelevant, as long as it is fast enough to spread out the sauce. The area to be covered depends only on how far you are from the center.
I agree, (with gasstationwithoutpumps), depending on how far you are away from the center of the pizza, will determine how much of the sauce is spread throughout the entire pizza. Meaning that if you are a little off from the center, the length that you are away will become the radius for the circle (pie*r^2) in the center that won’t be covered with sauce.
The rate of the spinning of the pizza would still be relevant. If thought about intuitively, the outer part of the pizza has more surface area. If the sauce and the arm is moving at a constant rate, then the pizza will have to spin slower when the arm is at the outer edge of the pizza.
The second case is harder to figure out since the rate at which the sauce is coming out has to be measured in surface area covered per second, or if taught in a more complex setting, how much (in volumes) sauce is distributed per second.
Another fun related rate question I ran into was asking about cassette tapes. Given the diameter of the initial roll, the thickness of the film, the radius of the plastic spinner, and the speed of the roll (radian/second), how long does it take to (re)wind a set of tape?