Activity 3.2.3.
A can of soda (at room temperature) is placed in a refrigerator at time \(t = 0\) (in minutes) and its temperature, \(F(t)\text{,}\) in degrees Fahrenheit, is computed at regular intervals. Based on the data, a model is formulated for the object’s temperature, given by
\begin{equation*}
F(t) = 42 + 30(0.95)^{t}\text{.}
\end{equation*}
(a)
Consider the simpler (parent) function \(p(t) = (0.95)^t\text{.}\) How do you expect the graph of this function to appear? How will it behave as time increases? Without using graphing technology, sketch a rough graph of \(p\) and write a sentence of explanation.
(b)
For the slightly more complicated function \(r(t) = 30 (0.95)^{t}\text{,}\) how do you expect this function to look in comparison to \(p\text{?}\) What is the long-range behavior of this function as \(t\) increases? Without using graphing technology, sketch a rough graph of \(r\) and write a sentence of explanation.
(c)
Finally, how do you expect the graph of \(F(t) = 42 + 30(0.95)^{t}\) to appear? Why? First sketch a rough graph without graphing technology, and then use technology to check your thinking and report an accurate, labeled graph on the axes provided below.
(d)
What is the temperature of the refrigerator? What is the room temperature of the surroundings outside the refrigerator? Why?
(e)
Determine the average rate of change of \(F\) on the intervals \([10,20]\text{,}\) \([20,30]\text{,}\) and \([30,40]\text{.}\) Write at least two careful sentences that explain the meaning of the values you found, including units, and discuss any overall trend in how the average rate of change is changing.
