Activity 1.2.2.
Consider a spherical tank of radius \(4\) m that is filling with water. Let \(V\) be the volume of water in the tank (in cubic meters) at a given time, 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 \(h\) according to the rule
\begin{equation*}
V = f(h) = \frac{\pi}{3} h^2(12-h)\text{.}
\end{equation*}
(a)
What values of \(h\) make sense to consider in the context of this function? What values of \(V\) make sense in the same context?
(b)
What is the domain of the function \(f\) in the context of the spherical tank? Why? What is the corresponding codomain? Why?
(c)
Determine and interpret (with appropriate units) the values \(f(2)\text{,}\) \(f(4)\text{,}\) and \(f(8)\text{.}\) What is important about the value of \(f(8)\text{?}\)
(d)
Consider the claim: “since \(f(9) = \frac{\pi}{3} 9^2(12-9) = 81\pi \approx 254.47\text{,}\) when the water is \(9\) meters deep, there is about \(254.47\) cubic meters of water in the tank”. Is this claim valid? Why or why not? Further, does it make sense to observe that “\(f(13) = -\frac{169\pi}{3}\)”? Why or why not?
