Activity 1.7.4.
During a major rainstorm, the rainfall at Gerald R. Ford Airport is measured on a frequent basis for a \(10\)-hour period of time. The following function \(g\) models the rate, \(R\text{,}\) at which the rain falls (in cm/hr) on the time interval \(t = 0\) to \(t = 10\text{:}\)
\begin{equation*}
R = g(t) = \frac{4}{t+2} + 1\text{.}
\end{equation*}
(a)
Compute \(g(3)\) and write a complete sentence to explain its meaning in the given context, including units.
(b)
Compute the average rate of change of \(g\) on the time interval \([3,5]\) and write two careful complete sentences to explain the meaning of this value in the context of the problem, including units. Explicitly address what the value you compute tells you about how rain is falling over a certain time interval, and what you should expect as time goes on.
(c)
Plot the function \(y = g(t)\) using a computational device. On the domain \([0,10]\text{,}\) what is the corresponding range of \(g\text{?}\) Why does the function \(g\) have an inverse function?
(d)
Determine \(g^{-1} \left( \frac{9}{5} \right)\) and write a complete sentence to explain its meaning in the given context.
(e)
According to the model \(g\text{,}\) is there ever a time during the storm that the rain falls at a rate of exactly \(1\) centimeter per hour? Why or why not? Provide an algebraic justification for your answer.
