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Problem 70 Hard Difficulty

Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?

Answer

Point $A$ is an inflection point and is significant because the rate of increase in depth changes from increasing to decreasing (depth still increasing but at a slowing rate).

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Video Transcript

So coffee is being poured into the moat shown in the figure. So the figure is right here and what we're being asked immeasurable, impertinent time. And we're being asked to sketch a graph. Ah, rough graph of the depth of the coffee in the mug at the function of time and account for the shape of the graph in terms of Khan. Captain. So what is the significance of this inflection point? So this is a kind of a rough sketch of the pitch to coffee that's in a textbook, but you can see that it is very wide at the bottom and then gets narrow and then gets right to get so they're just kind of a general idea of what graph looks like. So this is going to be the exact that's going to depth. So they get sick, and then the Y axis will mean the Y axis will be dead in the exact it will be time. So when we first begin point coffee at a constant rate, the debt won't The death won't be ah, increasing. It's much quicker this wide, and this takes it takes more coffee to fill up so's less of being achieved, and then it's going to sort of increase as they get into the more narrow range. It gets more into the narrow aged increases really fast, and then it gets wide again and sort of does the opposite, and that is what happened to forget. And so we have an inflection points. But this is Kong cave up, and then we have can't keep down the inflection point somewhere in the middle. And so the significance of the inflection point is that at this point, the coffee eyes being poured faster at any other point. So it is being filled up. The the height or the depth of the coffee in the monk is increasing at a faster rate than any other point in the coffee mug. And that's really the important. That's really the interpretation for the inflection point. And that's really what widen what's really need about inflection. Point of telling us about the fastest rate at which it is, um, tells us a lot about different situation in this term tells us that there rated the coffees. The depth of the company is increasing at a factory than any other point