Preview Activity 3.6.1.
In each of the following situations, determine the exact value of the unknown quantity that is identified.
(a)
The temperature of a warming object in an oven is given by \(F(t) = 275 - 203e^{-kt}\text{,}\) and we know that the object’s temperature after \(20\) minutes is \(F(20) = 101\text{.}\) Determine the exact value of \(k\text{.}\)
(b)
The temperature of a cooling object in a refrigerator is modeled by \(F(t) = a + 37.4e^{-0.05t}\text{,}\) and the temperature of the refrigerator is \(39.8^\circ\text{.}\) By thinking about the long-term behavior of \(e^{-0.05t}\) and the long-term behavior of the object’s temperature, determine the exact value of \(a\text{.}\)
(c)
Later in this section, we’ll learn that one model for how a population grows over time can be given by a function of the form
\begin{equation*}
P(t) = \frac{A}{1 + Me^{-kt}}\text{.}
\end{equation*}
Models of this form lead naturally to equations that have structure like
\begin{equation*}
3 = \frac{10}{1+x}.
\end{equation*}
Solve the equation \(3 = \frac{10}{1+x}\) for the exact value of \(x\text{.}\)
(d)
Suppose that \(y = a + be^{-kt}\text{.}\) Solve for \(t\) in terms of \(a\text{,}\) \(b\text{,}\) \(k\text{,}\) and \(y\text{.}\) What does this new equation represent?
