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...
$$\frac{dV}{dt}=c,\ c\in\Bbb R$$ $$\Rightarrow V=ct+C_1$$ 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.