The figure shows six containers, each of which is filled from ...

Question

The figure shows six containers, each of which is filled from the top. Assume water is poured into the containers at a constant rate and each container is filled in 10 s. Assume also that the horizontal cross sections of the containers are always circles. Let h (t) be the depth of water in the container at time t, for 0 ≤ t ≤ 10 . <IMAGE>

d. For each container, where does h' (the derivative of h ) have an absolute maximum on [0 , 10]?

d. For each container, where does h' (the derivative of h ) have an absolute maximum on [0 , 10]?

Accepted Answer

Welcome back, everyone. In this problem, a chemical is poured into cylindrical and conical flasks at a constant rate. It takes 8 seconds to fill each flask to the brim. If D of T represents the depth of the chemical at any time T, where T is greater than or equal to 0 but less than or equal to 8, for which flask does D reach an absolute maximum on the interval open bracket 08 closed bracket. Here we have images of our cylindrical and conical flask. Well, to figure out which one reach reaches an absolute maximum on the interval, let's try to understand how. Our depth changes for each of our flasks, OK? So let's start with the cylindrical flask. Cylindrical flask, OK? Now what do we know? What do we think is gonna happen as we pour our chemical into the flask? Well, since the cross section of the cylindrical flask, OK, since its cross section does not change as the chemical is poured at a constant rate, then that means the depth of the chemical increases steadily. In other words, if we were to put the function D of T on a graph, then it would be a linear function. It would look something like this. So we have the. Uh T as our independent variable, D as our dependent variable, and then it would increase steadily. Now, because it's a linear function, then we know that we could write. Day of tea As a function Y being equal to MX plus B, OK? That's the form of every linear function where X represents our variable. So in other words, we could actually write this as D of T equals M T + B rather, where M represents the slope and B represents a constant. Now, if we differentiate D. Then its derivative would be equal to m when we differentiate it with respect to T, and that is the constant and the slope of the linear function itself. Thus, for the cylindrical flask, there is no absolute maximum for D because it's a constant value between T being greater than R equal to 0 but less than R equal to 8. Now is that the same thing for our conica flask? Let's take a better look. Now for a conical flask, you might be able to observe that its cross section decreases from the bottom, OK, going up to the top. So here this would be where our cross section would be, so it's increasing here, right? And because it's increasing, because it's increasing or decreasing from the bottom to the top, that means when the chemical is poured at a constant rate, the depth, the increase in the depth is slower at first and goes faster as the chemical becomes closer to the brim. In other words, if we were to draw a graph here for whole or a depth is changing with respect to time, it would probably look something like that, OK? And now recall that a tangent line to this curve describes the slope of D at a certain time T. So that means when the tangent lines are drawn on the curve, OK, let's say we were to draw some tangent lines uh here. Oops, it's not a tangent line. So we draw it here, here, or here, OK. Then That means As they are drawn, it becomes more evident that the tangent lines get steeper as T increases, indicating indicating an increase in the rate of change in depth. So that means that the absolute maximum ford on the closed interval 08 is at T equals 8, where the tangent line is the steepest, OK. So the absolute maximum. Absolute maximum for the prime. Is at T equals 8, where our tangent line is steepest. Thus, only the conical flask has an absolute maximum ford which can be found at T equals 8. Thanks a lot for watching everyone. I hope this video helped.

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