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 = A _{0}*b^{t/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 = A _{0}*b^{t/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 100^{0 }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 5^{0 }C. If the temperature of the soup was 60^{0 }C after 10 minutes, how many minutes will it take for the temperature of the soup to reach a room temperature of 20^{0} 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 20^{0 }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 20^{0 }C, when time t = 33.77 minutes or about 34 minutes.

**Problem 2**: A 40^{0 }F roast is put into an oven that is set to bake at 350^{0 }F. After 2 hours, the temperature of the roast is 125^{0 }F. The roast is considered done when its internal temperature reaches 165^{0 }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 70^{0 }F room to cool, the temperature of a cake was 180^{0 }F. Three minutes later the temperature of the cake dropped to 160^{0 }F.

(a) What is the temperature of the cake after 20 minutes? (A: 98.87^{0 }F or about 99^{0 }F)

(b) How many minutes will take for the cake to cool to 90^{0 }F? (A: 25.49 minutes or about 26 minutes)

**Exercise 2**: The temperature of a very small metal bar was 30^{0 }C when it was dropped into a large barrel of hot water having a 75^{0 }C temperature. After 1 second, the temperature of the bar was 31^{0 }C.

(a) How long will it take for the temperature of the bar to reach 70^{0 }C? (A: 97.77 seconds or about 98 seconds)

(b) How long will it take for the temperature of the bar to reach 74^{0 }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)

**Comments:**

• 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 = Pe ^{rt}**. The post will assume that the reader has no understanding of the limit concept in calculus.