Activity 3.2.4.
A potato initially at room temperature (\(68^\circ\)) is placed in an oven (at \(350^\circ\)) at time \(t = 0\text{.}\) It is known that the potato’s temperature at time \(t\) is given by the function \(F(t) = a - b(0.98)^t\) for some positive constants \(a\) and \(b\text{,}\) where \(F\) is measured in degrees Fahrenheit and \(t\) is time in minutes.
(a)
What is the numerical value of \(F(0)\text{?}\) What does this tell you about the value of \(a - b\text{?}\)
(b)
Based on the context of the problem, what should be the long-range behavior of the function \(F(t)\text{?}\) Use this fact along with the behavior of \((0.98)^t\) to determine the value of \(a\text{.}\) Write a sentence to explain your thinking.
(c)
What is the value of \(b\text{?}\) Why?
(d)
Check your work above by plotting the function \(F\) using graphing technology in an appropriate window. Record your results on the axes provided below, labeling the scale on the axes. Then, use the graph to estimate the time at which the potato’s temperature reaches \(325\) degrees.
(e)
How can we view the function \(F(t) = a - b(0.98)^t\) as a transformation of the parent function \(f(t) = (0.98)^t\text{?}\) Explain.
