# pouring water into containers at a constant rate

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...

• By “decanter” you mean something like this? – gen-z ready to perish Dec 29 '17 at 6:08
• @ 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. – 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.
• 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 $f(t)$... – space Dec 30 '17 at 9:55
• Is $l$ your radius? – Harry Alli Dec 30 '17 at 11:52