What roles do the parameters \(a\text{,}\)\(k\text{,}\) and \(c\) play in how the function \(F(t) = c + ae^{-kt}\) models the temperature of an object that is cooling or warming in its surroundings?

How can we use an exponential function to more realistically model a population whose growth levels off?

We’ve seen that exponential functions can be used to model several different important phenomena, such as the growth of money due to continuously compounded interest, the decay of radioactive quanitities, and the temperature of an object that is cooling or warming due to its surroundings. From initial work with functions of the form \(f(t) = ab^t\) where \(b \gt 0\) and \(b \ne 1\text{,}\) we found that shifted exponential functions of form \(g(t) = ab^t + c\) are also important. Moreover, the special base \(e\) allows us to represent all of these functions through horizontal scaling by writing

where \(k\) is the constant such that \(e^k = b\text{.}\) Functions of the form of Equation (3.6.1) are either always increasing or always decreasing, always have the same concavity, are defined on the set of all real numbers, and have their range as the set of all real numbers greater than \(c\) or all real numbers less than \(c\text{.}\) In whatever setting we are using a model of this form, the crucial task is to identify the values of \(a\text{,}\)\(k\text{,}\) and \(c\text{;}\) that endeavor is the focus of this section.

We have also begun to see the important role that logarithms play in work with exponential models. The natural logarithm is the inverse of the natural exponential function and satisfies the important rule that \(\ln(b^k) = k\ln (b)\text{.}\) This rule enables us to solve equations with the structure \(a^k = b\) for \(k\) in the context where \(a\) and \(b\) are known but \(k\) is not. Indeed, we can first take the natural log of both sides of the equation to get

from which it follows that \(k \ln(a) = \ln(b)\text{,}\) and therefore

\begin{equation*}
k = \frac{\ln(b)}{\ln(a)}\text{.}
\end{equation*}

Finding \(k\) is often central to determining an exponential model, and logarithms make finding the exact value of \(k\) possible.

In Preview Activity 3.6.1, we revisit some key algebraic ideas with exponential and logarithmic equations in preparation for using these concepts in models for temperature and population.

Preview Activity3.6.1.

In each of the following situations, determine the exact value of the unknown quantity that is identified.

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 \(k\text{.}\)

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 \(a\text{.}\)

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

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?

Subsection3.6.1Newton’s Law of Cooling revisited

In Section 3.2, we learned that Newton’s Law of Cooling, which states that an object’s temperature changes at a rate proportional to the difference between its own temperature and the surrounding temperature, results in the object’s temperature being modeled by functions of the form \(F(t) = ab^t + c\text{.}\) In light of our subsequent work in Section 3.3 with the natural base \(e\text{,}\) as well as the fact that \(0 \lt b \lt 1\) in this model, we know that Newton’s Law of Cooling implies that the object’s temperature is modeled by a function of the form

for some constants \(a\text{,}\)\(c\text{,}\) and \(k\text{,}\) where \(k \gt 0\text{.}\)

From Equation (3.6.3), we can determine several different characteristics of how the constants \(a\text{,}\)\(b\text{,}\) and \(k\) are connected to the behavior of \(F\) by thinking about what happens at \(t = 0\text{,}\) at one additional value of \(t\text{,}\) and as \(t\) increases without bound. In particular, note that \(e^{-kt}\) will tend to \(0\) as \(t\) increases without bound.

Modeling temperature with Newton’s Law of Cooling.

For the function \(F(t) = ae^{-kt} + c\) that models the temperature of a cooling or warming object, the constants \(a\text{,}\)\(c\text{,}\) and \(k\) play the following roles. Note that \(k \gt 0\text{.}\)

Since \(e^{-kt}\) tends to \(0\) as \(t\) increases without bound, \(F(t)\) tends to \(c\) as \(t\) increases without bound, and thus \(c\) represents the temperature of the object’s surroundings.

Since \(e^0 = 1\text{,}\)\(F(0) = a + c\text{,}\) and thus the object’s initial temperature is \(a + c\text{.}\) Said differently, \(a\) is the difference between the object’s initial temperature and the temperature of the surroundings.

Once we know the values of \(a\) and \(c\text{,}\) the value of \(k\) is determined by knowing the value of the temperature function \(F(t)\) at one nonzero value of \(t\text{.}\)

Activity3.6.2.

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.

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.

Determine the exact time the object’s temperature is \(42.4^\circ\text{.}\) Clearly show your algebraic work and thinking.

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?

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?

Subsection3.6.2A more realistic model for population growth

If we assume that a population grows at a rate that is proportionate to the size of the population, it follows that the population grows exponentially according to the model

\begin{equation*}
P(t) = Ae^{kt}
\end{equation*}

where \(A\) is the initial population and \(k\) is tied to the rate at which the population grows. Since \(k \gt 0\text{,}\) we know that \(e^{kt}\) is an always increasing, always concave up function that grows without bound. While \(P(t) = Ae^{kt}\) may be a reasonable model for how a population grows when it is relatively small, because the function grows without bound as time increases, it can’t be a realistic long-term representation of what happens in reality. Indeed, whether it is the number of fish who can survive in a lake, the number of cells in a petri dish, or the number of human beings on earth, the size of the surroundings and the limitations of resources will keep the population from being able to grow without bound.

In light of these observations, a different model is needed for population, one that grows exponentially at first, but that levels off later. Calculus can be used to develop such a model, and the resulting function is usually called the logistic function, which has form

where \(A\text{,}\)\(M\text{,}\) and \(k\) are positive constants. Since \(k \gt 0\text{,}\) it follows that \(e^{-kt} \to 0\) as \(t\) increases without bound, and thus the denominator of \(P\) approaches \(1\) as time goes on. Thus, we observe that \(P(t)\) tends to \(A\) as \(t\) increases without bound. We sometimes refer to \(A\) as the carrying capacity of the population.

Activity3.6.3.

In Desmos, define \(P(t) = \frac{A}{1 + Me^{-kt}}\) and accept sliders for \(A\text{,}\)\(M\text{,}\) and \(k\text{.}\) Set the slider ranges for these parameters as follows: \(0.01 \le A \le 10\text{;}\)\(0.01 \le M \le 10\text{;}\)\(0.01 \le k \le 5\text{.}\)

Sketch a typical graph of \(P(t)\) on the axes provided and write several sentences to explain the effects of \(A\text{,}\)\(M\text{,}\) and \(k\) on the graph of \(P\text{.}\)

On a typical logistic graph, where does it appear that the population is growing most rapidly? How is this value connected to the carrying capacity, \(A\text{?}\)

How does the function \(1 + Me^{-kt}\) behave as \(t\) decreases without bound? What is the algebraic reason that this occurs?

Use your Desmos worksheet to find a logistic function \(P\) that has the following properties: \(P(0) = 2\text{,}\)\(P(2) = 4\text{,}\) and \(P(t)\) approaches \(9\) as \(t\) increases without bound. What are the approximate values of \(A\text{,}\)\(M\text{,}\) and \(k\) that make the function \(P\) fit these criteria?

Activity3.6.4.

Suppose that a population of animals (measured in thousands) that lives on an island is known to grow according to the logistic model, where \(t\) is measured in years. We know the following information: \(P(0) = 2.45\text{,}\)\(P(3) = 4.52\text{,}\) and as \(t\) increases without bound, \(P(t)\) approaches \(11.7\text{.}\)

Determine the exact values of \(A\text{,}\)\(M\text{,}\) and \(k\) in the logistic model

Plot your model from (a) and check that its values match the desired characteristics. Then, compute the average rate of change of \(P\) on the intervals \([0,2]\text{,}\)\([2,4]\text{,}\)\([4,6]\text{,}\) and \([6,8]\text{.}\) What is the meaning (with units) of the values you’ve found? How is the population growing on these intervals?

Find the exact time value when the population will be \(10\) (thousand). Show your algebraic work and thinking.

Subsection3.6.3Summary

When a function of form \(F(t) = c + ae^{-kt}\) models the temperature of an object that is cooling or warming in its surroundings, the temperature of the surroundings is \(c\) because \(e^{-kt} \to 0\) as time goes on, the object’s initial temperature is \(a+c\text{,}\) and the constant \(k\) is connected to how rapidly the object’s temperature changes. Once \(a\) and \(c\) are known, the constant \(k\) can be determined by knowing the temperature at one additional time, \(t\text{.}\)

Because the exponential function \(P(t) = Ae^{kt}\) grows without bound as \(t\) increases, such a function is not a realistic model of a population that we expect to level off as time goes on. The logistic function

more appropriately models a population that grows roughly exponentially when \(P\) is small but whose size levels off as it approaches the carrying capacity of the surrounding environment, which is the value of the constant \(A\text{.}\)

Exercises3.6.4Exercises

1.

Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of cooling. If the coffee has a temperature of \(210\) degrees Fahrenheit when freshly poured, and \(1.5\) minutes later has cooled to \(195\) degrees in a room at \(78\) degrees, determine when the coffee reaches a temperature of \(155\) degrees.

The coffee will reach a temperature of \(155\) degrees in minutes.

2.

The total number of people infected with a virus often grows like a logistic curve. Suppose that 25 people originally have the virus, and that in the early stages of the virus (with time, \(t\text{,}\) measured in weeks), the number of people infected is increasing exponentially with \(k=1.8\text{.}\) It is estimated that, in the long run, approximately 7250 people become infected.

(a) Use this information to find a logistic function to model this situation.

\(P =\)

(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point?

vertical coordinate =

In Exercise 3.6.4.3 below, use the following structure/formula for \(N(t)\text{:}\)\(N(t)=\frac{L}{1+Ab^{-kt}}\text{.}\) In particular, note that when the instructions say “find \(A\)”, this use of “\(A\)” is not in reference to carrying capacity.

3.

The town of Sickville, with a population of 9310 is exposed to the Blue Moon Virus, against which there is no immunity. The number of people infected when the virus is detected is 30. Suppose the number of infections grows logistically, with \(k = 0.18\text{.}\)

Find \(A\text{.}\)

Find the formula for the number of people infected after \(t\) days.

\(N(t) =\)

Find the number of people infected after 30 days.

4.

A glass filled with ice and water is set on a table in a climate-controlled room with constant temperature of \(71^\circ\) Fahrenheit. A temperature probe is placed in the glass, and we find that the following temperatures are recorded (at time \(t\) in minutes).

\(t\)

\(0\)

\(20\)

\(F(t)\)

\(34.2\)

\(41.7\)

Make a rough sketch of how you think the temperature graph should appear. Is the temperature function always increasing? always decreasing? always concave up? always concave down? what’s its long-range behavior?

By describing \(F\) as a transformation of \(e^t\text{,}\) explain why a function of form \(F(t) = c - ae^{-kt}\text{,}\) where \(a\text{,}\)\(c\text{,}\) and \(k\) are positive constants is an appropriate model for how we expect the temperature function to behave.

Use the given information to determine the exact values of \(a\text{,}\)\(c\text{,}\) and \(k\) in the model \(F(t) = c - ae^{-kt}\text{.}\)

Determine the exact time when the water’s temperature is \(60^\circ\text{.}\)

5.

A popular cruise ship sets sail in the Gulf of Mexico with \(5000\) passengers and crew on board. Unfortunately, a five family members who board the ship are carrying a highly contagious virus. After interacting with many other passengers in the first few hours of the cruise, all five of them get very sick.

Let \(S(t)\) be the number of people who have acquired the virus \(t\) days after the ship has left port. It turns out that a logistic function is a good model for \(S\text{,}\) and thus we assume that

for some positive constants \(A\text{,}\)\(M\text{,}\) and \(k\text{.}\) Suppose that after \(1\) day, \(20\) people have gotten the virus.

Recall we know that \(S(0) = 5\) and \(S(1) = 20\text{.}\) In addition, assume that \(5000\) is the number of people who will eventually get sick. Use this information determine the exact values of \(A\text{,}\)\(M\text{,}\) and \(k\) in the logistic model.

How many days will it take for \(4000\) of the people on the cruise ship to have acquired the virus?

Compute the average rate of change of \(S\) on the intervals \([1,2]\text{,}\)\([3,4]\text{,}\) and \([5,7]\text{.}\) What is the meaning of each of these values (with units) in the context of the question, and what trend(s) do you observe in these average rates of change?

6.

A closed tank with an inflow and outflow contains a \(100\) liters of saltwater solution. Let the amount of salt in the tank at time \(t\) (in minutes) be given by the function \(A(t)\text{,}\) whose output is measured in grams. At time \(t = 0\) there is an initial amount of salt present in the tank, and the inflow line also carries a saltwater mixture to the tank at a fixed rate; the outflow occurs at the same rate and carries a perfectly mixed solution out of the tank. Because of these conditions, the volume of solution in the tank stays fixed over time, but the amount of salt possibly changes.

It turns out that the problem of determining the amount of salt in the tank at time \(t\) is similar to the problem of determining the temperature of a warming or cooling object, and that the function \(A(t)\) has form

\begin{equation*}
A(t) = ae^{-kt} + c
\end{equation*}

for constants \(a\text{,}\)\(c\text{,}\) and \(k\text{.}\) Suppose that for a particular set of conditions, we know that

Again, \(A(t)\) measures the amount of salt in the tank after \(t\) minutes.

How much salt is in the tank initially?

In the long run, how much salt do we expect to eventually be in the tank?

At what exact time are there exactly \(500\) grams of salt present in the tank?

Can you determine the concentration of the solution that is being delivered by the inflow to the tank? If yes, explain why and determine this value. If not, explain why that information cannot be found without additional data.