For each of the following, we will be pouring water at a constant rate into a container, and we let $f(t)$ denote the height of the water at time $t$. Sketch the graph of $f$ if the container is the following, paying particular attention to the shape and concavity of your graph: a) cylinder b) cone c) a decanter

I actually don't know where to start here. I was thinking that if there is a function $V(t)$ that represented the amount of water poured at time $t$, then for part (a) $\frac{V(t)}{\pi r^2} = f(t)$ for a cylinder with radius $r$. Also, the "constant rate" thing seems to tell me that $f''(t)=0$, but I don't know what next...

  • $\begingroup$ By “decanter” you mean something like this? $\endgroup$ – gen-z ready to perish Dec 29 '17 at 6:08
  • $\begingroup$ @ Chase Ryan Taylor yes. It is described as "narrow at the bottom, widens in the middle, then narrows towards the top" in the note below the problem. $\endgroup$ – space Dec 29 '17 at 7:54

$$\frac{dV}{dt}=c,\ c\in\Bbb R$$ $$\Rightarrow V=ct+C_1$$ We know that when time is zero, there is no water in the tank, so $C_1=0$

For a cylinder, $$V=\pi r^2f(t) \Rightarrow f(t)=\frac{ct}{\pi r^2}$$ So, as a function of $t$, this is just a straight line.

A similar method can be used in the other scenarios.

However, an important thing to note in the cone case is that we cannot treat the radius as a constant, because this changes with the height. However, with some simple Pythagoras, one can rewrite the radius in terms of the height at a particular time.

  • $\begingroup$ Okay, so for the cone I got that $ct = \frac{1}{3}\pi f(t)l^2 - f^3(t)$, but I can't seem to isolate $f(t)$... $\endgroup$ – space Dec 30 '17 at 9:55
  • $\begingroup$ Is $l$ your radius? $\endgroup$ – Harry Alli Dec 30 '17 at 11:52
  • $\begingroup$ It is the slant height $\endgroup$ – space Dec 30 '17 at 13:44
  • $\begingroup$ Well, firstly, your slant height is a function of the height as well. It was easier with the cylinder example, but an important thing to note is that not all cones are similar. You can have fatter cones and skinnier cones, so the initial conditions of the vessel are important. So the equation is dependent on the initial ratio between height and radius. $\endgroup$ – Harry Alli Dec 30 '17 at 13:48

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