3.6 Modeling temperature and population

Introduction

Newton's Law of Cooling revisited

A more realistic model for population growth

Summary

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 $F(t) = 275 - 203e^{-kt}$, and we know that the object's temperature after $20$ minutes is $F(20) = 101$. Determine the exact value of $k$.

Use algebraic reasoning and your understanding of the physical situation to determine the exact values of $a$, $c$, and $k$ in the model $F(t) = ae^{-kt}+c$. Write at least one careful sentence to explain your thinking.

Determine the exact time the object's temperature is $42.4^\circ$. Clearly show your algebraic work and thinking.

In Desmos, enter the values you found for $a$, $c$, and $k$ in order to define the function $F$. Then, use Desmos to find the average rate of change of $F$ on the interval $[25,30]$. What is the meaning (with units) of this value?

If everything stayed the same except the value of $F(0)$, and instead $F(0) = 65$, would the value of $k$ be larger or smaller? Why?

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

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, $A$?

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: $P(0) = 2$, $P(2) = 4$, and $P(t)$ approaches $9$ as $t$ increases without bound. What are the approximate values of $A$, $M$, and $k$ that make the function $P$ fit these criteria?

Determine the exact values of $A$, $M$, and $k$ in the logistic model

. Clearly show your algebraic work and thinking.

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 $[0,2]$, $[2,4]$, $[4,6]$, and $[6,8]$. 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.

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 $e^t$, explain why a function of form $F(t) = c - ae^{-kt}$, where $a$, $c$, 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 $a$, $c$, and $k$ in the model $F(t) = c - ae^{-kt}$. - Determine the exact time when the water's temperature is $60^\circ$.

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 $S$, and thus we assume that

for some positive constants $A$, $M$, and $k$. Suppose that after $1$ day, $20$ people have gotten the virus.

• Recall we know that $S(0) = 5$ and $S(1) = 20$. In addition, assume that $5000$ is the number of people who will eventually get sick. Use this information determine the exact values of $A$, $M$, 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 $[1,2]$, $[3,4]$, and $[5,7]$. 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?

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 $A(t)$, 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

for constants $a$, $c$, and $k$. 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.