3.2 Modeling with exponential functions (continued)

Modeling temperature data

Summary

Consider the simpler (parent) function $p(t) = (0.95)^t$. 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.

For the slightly more complicated function $r(t) = 30 (0.95)^{t}$, how do you expect this function to look in comparison to $p$? 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.

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.

What is the temperature of the refrigerator? What is the room temperature of the surroundings outside the refrigerator? Why?

Determine the average rate of change of $F$ on the intervals $[10,20]$, $[20,30]$, and $[30,40]$. 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.

What is the numerical value of $F(0)$? What does this tell you about the value of $a - b$?

Based on the context of the problem, what should be the long-range behavior of the function $F(t)$? Use this fact along with the behavior of $(0.98)^t$ to determine the value of $a$. Write a sentence to explain your thinking.

What is the value of $b$? Why?

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 $180$ degrees.

How can we view the function $F(t) = a - b(0.98)^t$ as a transformation of the parent function $f(t) = (0.98)^t$? Explain.

A can of soda has been in a refrigerator for several days; the refrigerator has temperature $41^\circ$ Fahrenheit. Upon removal, the soda is placed on a kitchen table in a room with surrounding temperature $72^\circ$. Let $F(t)$ represent the soda's temperature in degrees Fahrenheit at time $t$ in minutes, where $t = 0$ corresponds to the time the can is removed from the refrigerator. We know from Newton's Law of Cooling that $F$ has form $F(t) = ab^t + c$ for some constants $a$, $b$, and $c$, where $0 \lt b \lt 1$.

• What is the numerical value of the soda's initial temperature? What is the value of $F(0)$ in terms of $a$, $b$, and $c$? What do these two observations tell us? - What is the numerical value of the soda's long-term temperature? What is the long-term value of $F(t)$ in terms of $a$, $b$, and $c$? What do these two observations tell us? - Using your work in (a) and (b), determine the numerical values of $a$ and $c$. - Suppose it can be determined that $b = 0.931$. What is the soda's temperature after $10$ minutes?

Consider the graphs of the following four functions $p$, $q$, $r$, and $s$. Each is a shifted exponential function of the form $ab^t + c$.

For each function $p$, $q$, $r$, and $s$, determine - whether $a \gt 0$ or $a \lt 0$; - whether $0 \lt b \lt 1$ or $b \gt 1$; - whether $c \gt 0$, $c = 0$, or $c \lt 0$; and - the range of the function in terms of $c$.

A cup of coffee has its temperature, $C(t)$, measured in degrees Celsius. When poured outdoors on a cold morning, its temperature is $C(0) = 95$. Ten minutes later, $C(10) = 80$. If the surrounding temperature outside is $0^\circ$ Celsius, find a formula for a function $C(t)$ that models the coffee's temperature at time $t$.

In addition, recall that we can convert between Celsius and Fahrenheit according to the equations $F = \frac{9}{5}C + 32$ and $C = \frac{5}{9}(F-32)$. Use this information to also find a formula for $F(t)$, the coffee's Fahrenheit temperature at time $t$. What is similar and what is different regarding the functions $C(t)$ and $F(t)$?