Activity 3.6.2.
A can of soda is initially at room temperature, \(72.3^\circ\) Fahrenheit, and at time \(t = 0\) is placed in a refrigerator set at \(37.7^\circ\text{.}\) In addition, we know that after \(30\) minutes, the soda’s temperature has dropped to \(59.5^\circ\text{.}\) Let \(F(t)\) represent the temperature of the soda in degrees Fahrenheit at time \(t\) in minutes.
(a)
Use algebraic reasoning and your understanding of the physical situation to determine the exact values of \(a\text{,}\) \(c\text{,}\) and \(k\) in the model \(F(t) = ae^{-kt}+c\text{.}\) Write at least one careful sentence to explain your thinking.
(b)
Determine the exact time the object’s temperature is \(42.4^\circ\text{.}\) Clearly show your algebraic work and thinking.
(c)
In Desmos, enter the values you found for \(a\text{,}\) \(c\text{,}\) and \(k\) in order to define the function \(F\text{.}\) Then, use Desmos to find the average rate of change of \(F\) on the interval \([25,30]\text{.}\) What is the meaning (with units) of this value?
(d)
If everything stayed the same except the value of \(F(0)\text{,}\) and instead \(F(0) = 65\text{,}\) would the value of \(k\) be larger or smaller? Why?
