## Solving Newton’s Law of Cooling/Heating Problems without Differential Calculus

My last post discussed how to find an exponential growth/decay equation that expresses a relationship between two variables by first constructing a table of data-pairs to better understand and derive the fundamental grow/decay equation A = A0*bt/k. Because the content of this post depends on the concepts developed in my last post, I strongly suggest that you read that post before continuing.

This post shows how to solve Newton’s law of cooling and heating problems without any understanding of differential calculus, which makes this post different from descriptions found in differential calculus text books. Newton’s Law of Cooling describes the relationship between the temperature of an object and time t when the object is placed in an environment where the ambient (or surrounding) temperature is maintained at a constant temperature. Newton’s law of cooling and heating is described as follows:

(a) If the initial temperature of the object equals the ambient temperature, the temperature of the object remains constant as time t increases.

(b) If the initial temperature of the object is greater than the ambient temperature, the object cools and its temperature exponentially and asymptotically approaches the ambient temperature as time t increases.

(c) If the initial temperature of the object is less than the ambient temperature, the object heats up and its temperature exponentially and asymptotically approaches the ambient temperature as time t increases.

I will use two familiar cooling/heating problems to illustrate how the table data-pair approach can be applied to solve a Newton’s law of cooling or heating problem. The key step in solving a cooling/heating problem is to carefully read the problem and then apply what Newton tells us about cooling and heating to create a rough sketch of the growth/decay graph of the model with key points labeled. Even if you don’t know the equation of the graph, the rough sketch will enable you to determine the parameters of the growth/decay equation. From this rough sketch, recognize that the graph is just the result of a vertical translation of an exponential decay graph in the form A = A0*bt/k. (In view of what Newton tells us about cooling and heating, the rough graph makes perfect sense to students.)

Problem 1: A pot of boiling soup is put into a sink filled with cold water. The temperature of the soup was 1000 C when it was first put into the sink. By adding ice and stirring the water, the temperature of the water was maintained at a constant temperature of 50 C. If the temperature of the soup was 600 C after 10 minutes, how many minutes will it take for the temperature of the soup to reach a room temperature of 200 C?

Solution: Refer to graphs A and B below where x = time t in minutes and y = the temperature of the soup in degrees Celsius. Similar to graph A, first draw a rough sketch of the model with key points labeled. Recognizing that graph A is just the result of a 5 unit vertical translation of an exponential decay graph, use the information from the first rough sketch to draw a rough sketch of the exponential decay graph with key points labeled, similar to graph B. Now use the key points on the sketch of graph B to find the equation of graph B, and then apply the equation transformation rules to find the equation of graph A. To find out how many minutes it will take for the temperature of the soup to reach 200 C, use a computer graphing program to find the intersection point of the graphs y = 20 and y = 95(55/95)x/10 + 5. Graph A tells us the temperature of the soup equals 200 C, when time t = 33.77 minutes or about 34 minutes.  Problem 2: A 400 F roast is put into an oven that is set to bake at 3500 F. After 2 hours, the temperature of the roast is 1250 F. The roast is considered done when its internal temperature reaches 1650 F. How many hours well it take to cook the roast?

Solution: Refer to graphs C and D below where x = time t in minutes and y = the temperature of the roast in degrees Fahrenheit. The strategy is to first draw a rough sketch of the model with key points labeled; similar to graph C below. Recognizing that graph C is the result of a 350 unit vertical translation of an exponential decay graph that was reflected over the x-axis, use the information from the sketch of graph C to draw a rough sketch of the flipped exponential decay graph with key points labeled, similar to graph D. Now use the key points on the sketch of graph D to find the equation of Graph D, and then apply the equation transformation rules to find the equation of graph C. To find out how many hours it will take to cook the roast, use a computer graphing program to find the intersection point of the graphs y = 165 and y = -310(225/310)x/2 + 350. Graph C tells us that it will take 3.22 hours or about 3 hours and 13 minutes to cook the roast.  Here are four exercises that you can give to your students. The solutions are provided. (See my third comment below.) You or your students shouldn’t be too disappointed if you fail to correctly solve all four exercises on your first attempt.

Exercise 1: When first removed from an oven and placed in a 700 F room to cool, the temperature of a cake was 1800 F. Three minutes later the temperature of the cake dropped to 1600 F.

(a) What is the temperature of the cake after 20 minutes? (A: 98.870 F or about 990 F)
(b) How many minutes will take for the cake to cool to 900 F? (A: 25.49 minutes or about 26 minutes)

Exercise 2: The temperature of a very small metal bar was 300 C when it was dropped into a large barrel of hot water having a 750 C temperature. After 1 second, the temperature of the bar was 310 C.

(a) How long will it take for the temperature of the bar to reach 700 C? (A: 97.77 seconds or about 98 seconds)
(b) How long will it take for the temperature of the bar to reach 740 C? (A: 169.39 seconds or about 170 seconds)

Exercise 3: Find the equation of graph A below.  (A: y = 40(1/5)x/10 + 30)

Exercise 4: Find the equation of graph B below. (A: y = -90(2/3)x/5 + 160) • In the two sample problems above, the final step in the solution involved finding the intersection point of two graphs. This gives us the solution from a geometric point of view. The solution from an algebraic point of view involves log functions which would enable you to find the solution faster. As I mentioned in previous posts, whenever possible, solutions to problems should be understood from both an algebraic and geometric point of view.

• Solving exponential growth and decay problems naturally leads to a need to understand logarithms and log functions.

• All modern physicists know that the equations they discovered can only give us an approximation of how nature’s laws work. In reference to problem (2) above, if we conducted an experiment with a roast by measuring its internal temperature at various points in time, we would find a discrepancy between the experimental results and the predicted results. No matter how accurately we measure the internal temperature of the roast and time, the errors can’t be taken out of the experimental observations. We can only say that the interval temperate of the roast at some specific point in time lies in an area of uncertainly which is the area under a probability distribution curve. This is why least-squares regression equations are used to describe the relationship between two variables.

• I have used the handout Newton’s Law of Cooling with college algebra and pre-calculus students, and with more advanced students that I tutor. To download the free student and teacher versions of the handout, go to mathteachersresource.com/instructional-content. There are other free handouts on properties of exponents, properties of logarithms, solving exponential/logarithmic equations, and logarithmic base conversion.

• Using the approach presented in my last post and this post, I believe it’s possible to teach how to solve exponential growth/decay problems to younger mathematically capable students. From my own experience, students find these types of problems interesting and practical.

• All graphs in this post were created with my program Basic Trig Functions. I designed the program to make it easy for teachers to create content for their own courses.

My next post will discuss the derivation of the formula for the future value of an investment when interest is compound continuously, FV = Pert. The post will assume that the reader has no understanding of the limit concept in calculus.

## Exponential Growth and Decay from a Data-Pairs Approach My last two posts discussed the mathematics of linear growth and decay. If you have not read those posts, you might find it helpful to read them before continuing. This post focuses on finding an exponential equation that expresses a relationship between two variables by first constructing a table of data-pairs to better understand the relationship and see the pattern in the relationship.

Most exponential growth/decay relationships involve a time variable t and the amount A of some quantity at time t. Amount could be the current value of an investment account, population of a city, remaining kilograms of radioactive material, assessed value of a truck, etc. The text box and observations below explain how and why the basic fundamental exponential growth/decay formula A = A0*bt/k works, and the role that the parameters A0, b, and k play in the equation. Periodic growth factor is another way to think of the base multiplier b. Some observations about A = A0*bt/k where b > 0:
• The point (0, A0) is the intercept on the vertical axis of the graph.
• Base multiplier b is a periodic growth or decay factor.
• If 0 < b < 1, the equation models exponential decay.
• If b > 1, the equation models exponential growth.
• Exponential growth/decay is about repeated multiplication by growth/decay factor b.
• A0 and any other point on the graph determines a unique exponential equation.
• If A0 is positive, the graph is above and asymptotic to the horizontal axis.
• If A0 is negative, the graph is below and asymptotic to the horizontal axis.

Most discussions about finding the equation of an exponential relationship don’t start by looking at data-pairs in a table. After only a couple of demonstrations of how to apply the data-pairs approach, students quickly develop the ability to find the three key parameters of an exponential growth/decay relationship. Exponential equations of the form A = A0*bt/k where base b is a rational number are much easier to comprehend than equations of the form A = A0*ekt where e is the irrational math constant = 2.718281828459045 . . .  . I will use four familiar math problems that involve an exponential relationship to illustrate the table data-pairs approach. In the comments section of this post, you will find an example that further clarifies my reason for expressing most exponential growth/decay equations as A = A0*bt/k where b is a rational number, and the reason that the solutions of population and radioactive growth/decay problems tend to be expressed in terms of base e only.

Problem 1: Consider a population of bacteria that is growing exponentially 50% every 4 hours and the current population is 60 bacteria. Let t = number of hours in the future and N = the number of bacteria after t hours.
(a) Find an equation that expresses N as a function of t.
(b) Find the population after 10 hours and 45 minutes ago.
(c) Express N as a function of t if the population is increasing 5% every 15 minutes.

The solution is given in the text box below. Problem solvers should carefully read the problem, create a table of data-pairs, determine the equation parameters, and then write the equation that models the problem situation. A companion exponential growth graph with a series of slope/rate triangles is provided to show the role that the equation parameters play in the relationship. Of course, the problem solver should always check the solution by using a computer graphing program to graph the equation.  Problem 2: Suppose a person invests \$10,000 in a CD that will earn interest at 6%/year and interest is compounded monthly. Let t = the number of years in the future and V = the value of the investment after t years.
(a) Express V as a function of t.
(b) Find the value of the investment after 10 years and 20 years.
(c) Express V as a function of t if interest is compounded 360 times per year. Problem 3: The half-life of a radioactive substance equals the time it takes (20 days, 149 years, 5,700 years, etc.) for the substance to lose half its mass. Consider a radioactive substance with a half-life of 60 days that currently has a 100 kg mass. Let time t = the number of days in the future and A = the mass of the remaining substance in kg at time t. Refer to table and companion graph below. (a) Find a formula that expresses A as a function of time t in days.
(b) Find the mass of the substance after 135 days.
(c) Find a formula for A(t) if the half-life = 6 hours instead of 60 days.

Problem 4: The two exponential growth/decay graphs along with key points on the graphs are shown below.
(a) For graph A: Write an equation that expresses y as a function of x.
(b) For graph B: Write an equation that expresses y as a function of x.  Here are four exercises that you can give to your students. The graphs are a mixture of linear and exponential growth/decay graphs. Using the points on the graph, find the equation of the graph. If you wish, remind them that they should first create a table of data-pairs. Let them do the exercises with a partner and then check their answers by using a computer to graph the equations. We want to create a save environment in which kids feel free to experiment and check their answers for understanding. It’s OK to make a mistake, just fix it. If the first attempt to fix a mistake fails, so what? Try again. This is how real people learn to do anything that is worthwhile. The solutions are given at the end of this post. • Consider the two mathematically equivalent equations below that model the population growth of a small town where t equals the number of years after 2010.

P = 5,200(1.08)t/4  and  P = 5,200e0.019240260t

The first equation immediately tells us the population of the town was 5,200 in 2010, and the population is increasing 8% every 4 years. The second equation tells us the population of the town was 5,200 in 2010, but by just inspecting the second equation, only God can figure out that the population is increasing 8% every four years. (Increasing 8% every 4 years is slightly less than increasing 2% every year.)

• It’s a snap to find the derivative of functions of the form y = Aekt. To find the derivative of functions of the form y = Abx/k where base b is a rational number requires a little more work. I suspect this is the reason that the solutions of population and radioactive grow/decay problems tend to be expressed in terms of base e only. From my point of view, this is not a sufficient reason to do so because converting an exponential function from one base to another base is a simple procedure. My free handout Logarithmic Base Conversion shows how to do this.

• All modern physicists know that the equations they discovered can only give us an approximation of how nature’s laws work. The brilliant physicist Richard Feynman, over and over again, stated this fundamental fact in his lectures and talks. In reference to problem (3) above, if we conducted an experiment with a radioactive material by measuring the remaining mass of the material at various points in time, we would find a discrepancy between the experimental results and the predicted results. No matter how accurately we measure mass and time, the errors can’t be taken out of the experimental observations. We can only say that the remaining mass of radioactive material at time t lies in an area of uncertainly which is the area under a probability distribution curve. This is why least-squares regression equations are used to describe the relationship between two variables.

• The formula for calculating the future value of an account after t years when interest is compounded continuously is FV = Pert where P = the principal and r = the annual interest rate expressed as a decimal. It’s impossible to express this relationship with a base that is a rational number. In a future post, I will give a derivation of this formula in a manner that does not require an understanding of concepts in calculus.

• I have used the handout, Introduction to Exponential Growth and Decay, with college algebra students, pre-calculus students, and as a review for more advanced students. To download the free student and teacher versions of the handout, go to mathteachersresource.com/instructional-content.html. There are other free handouts on properties of exponents, properties of logarithms, solving exponential/logarithmic equations, and logarithmic base conversion.

• All graphs in this post were created with my software, Basic Trig Functions. I designed the software to help teachers quickly make custom content for their classrooms. This software allows you to easily copy any graphic and then import it directly into a document (e.g. lesson plan, class handout, test) or further manipulate it in various graphic processing programs.

My next post will show how to solve Newton’s Law of Cooling problems without understanding differential calculus.

Solutions to exercises:
Graph A: y = -2x + 40
Graph B: y = 40*0.5x/2.5
Graph C: y = 20*1.5x/5
Graph D: y = x + 10