Activity 1.2.3.
Consider a spherical tank of radius \(4\) m that is completely full of water. Suppose that the tank is being drained by regulating an exit valve in such a way that the height of the water in the tank is always decreasing at a rate of \(0.5\) meters per minute. Let \(V\) be the volume of water in the tank (in cubic meters) at a given time \(t\) (in minutes), and \(h\) the depth of the water (in meters) at the same time. It can be shown using calculus that \(V\) is a function of \(t\) according to the model
\begin{equation*}
V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t)\text{.}
\end{equation*}
In addition, let \(h = q(t)\) be the function whose output is the depth of the water in the tank at time \(t\text{.}\)
(a)
What is the height of the water when \(t = 0\text{?}\) When \(t = 1\text{?}\) When \(t = 2\text{?}\) How long will it take the tank to completely drain? Why?
(b)
What is the domain of the model \(h = q(t)\text{?}\) What is the domain of the model \(V = p(t)\text{?}\)
(c)
How much water is in the tank when the tank is full? What is the range of the model \(h = q(t)\text{?}\) What is the range of the model \(V = p(t)\text{?}\)
(d)
We will frequently use a graphing utility to help us understand function behavior, and strongly recommend Desmos because it is intuitive, online, and free. To learn more about Desmos, see their outstanding online tutorials.
In this prepared Desmos worksheet, you can see how we enter the (abstract) function \(V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t)\text{,}\) as well as the corresponding graph the program generates. Make as many observations as you can about the model \(V = p(t)\text{.}\) You should discuss its shape and overall behavior, its domain, its range, and more.
(e)
How does the model \(V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t)\) differ from the abstract function \(y = r(x) = \frac{256\pi}{3} - \frac{\pi}{24} x^2(24-x)\text{?}\) In particular, how do the domain and range of the model differ from those of the abstract function, if at all?
(f)
How should the graph of the height function \(h = q(t)\) appear? Can you determine a formula for \(q\text{?}\) Explain your thinking.
