Precalculus Honors 3.4 Newton s Law of Cooling November 8, 2005 Mr. DeSalvo Isaac Newton (1642-1727) proposed that the temperature of a hot object decreases at a rate proportional to the difference between its temperature and the temperature of its surroundings. Similarly, a cold object heats up at a rate proportional to the temperature difference between the object and the temperature of its surroundings. In this section, we will explore this concept known as Newton s Law of Cooling. EXAMPLE Imagine that on a cold autumn day you pour yourself a hot cup of coffee and place it on a table. Originally, the temperature of the coffee was 200 F and the temperature of the room was 70 F. Let t n be the temperature of the coffee n minutes after you place it on the table. 1. Make a careful sketch of what you think the temperature (in F) would look like as a function of the time (in minutes). Label the vertical axes temperature ( F) and the horizontal axis time (minutes). Solution: 350 Temperature of a Cup of Coffee ( F) 300 250 200 150 100 50 Time (minutes) 5 10 15 2. What is the value of t 0? Solution: t 0 = 200 F. 3. Over the long run, what will happen to the temperature of the coffee? Solution: Over the long run, the temperature of the coffee will level off at 70 F.
Newton s Law of Cooling states: the temperature of a hot object decreases at a rate proportional to the difference between its temperature and the temperature of its surroundings. In other words, for your cup of coffee, t n+1 t n = k(t n 70) for some unknown constant k. 4. Suppose that one minute after you place the cup of coffee on the table, its temperature is 180 F. Use this information to determine the value of k. Solution: We know, by Newton s Law of Cooling, that, t n+1 t n = k(t n 70). We also know that t 0 = 200 and t 1 = 180, thus we have 180-200 = k(200 70). Solving for k, we obtain k = (180 200)/(200 70) = -20/130 = -2/13. 5. Based on your answer to question 4, you now have enough information to completely determine the recursive formula for a sequence T that will generate the successive terms that give the temperature of the coffee, n minutes after it is placed in a room whose temperature is 70 F. Solution: t 0 = 200;, t n+1 t n = (-2/13)(t n 70). 6. Rewrite the recursive equation from part (5) so that it expresses t n+1 in terms of t n. (this will put the equation in standard form like the drug concentration model.) Solution: All we need to do here is add t n to both sides and simplify. Doing so, we obtain : t 0 = 200;, t n+1 = (11/13)t n + 140/13. EXERCISE 1. Use your TI (or make a spreadsheet) to generate successive terms of the sequence T. (If using the TI, remember that you first type 200 and hit ENTER. Next, type (11/13)Ans + 140/13 and hit ENTER again. From here on, each time you hit ENTER you will get a successive term of the sequence. To see the long run behavior, you need to hit ENTER about 40 times.) 2. According to question 1, what is the equilibrium point of this model? Why does this make sense given the physical conditions of this problem situation? 3. Find the equilibrium point of this model algebraically. 4. Use the method of iteration to find an explicit formula for t n. Check your answer by generating the first five terms of sequence T both recursively and explicitly. 5. Use your TI (or make a spreadsheet) to find when the temperature of the coffee will reach 85 F. 6. Repeat question 5 using your explicit formula.
HOMEWORK QUESTIONS 1. You bake a yam for Mr. DeSalvo at 350, and when you remove it from the oven, you let the yam cool in your dorm room, which has a temperature of 68 F. After 10 minutes, the yam has cooled to 240 F. Let t n be the temperature of the yam n 10-minute periods after you take it out of the oven. (a) Write a complete recursive formula for sequence T using Newton s Law of Cooling. (b) Make a spreadsheet that generates and graphs sequence T. Use enough terms to show the long-term behavior. Save your work as yam SS and print it out. (c) Find an explicit formula for t n using the method of iteration. (d) What is the temperature of the yam 1 hour after you take it out of the oven? (e) When will the temperature of the yam be 75 F. 2. Newton s Law of Heating uses the same mathematical model as the law of cooling. Suppose you take some soup from the dining room and place it in your refrigerator. Later that evening you take the soup, which is now cooled to 36 F, and place it in an oven that has been preheated to 400 F. Five minutes later, the temperature of the soup is 120 F. (a) Let T be a sequence that gives the temperature of the soup over time. Write a recursive formula for the sequence T using Newton s Law of Cooling, which has now become Newton s Law of Heating. (b) Find the equilibrium point of this model. How do you know this? (c) Make a spreadsheet that generates and graphs sequence T. Use enough terms to show the long-term behavior. Save your work as heating soup SS and print it out. (d) Find an explicit formula for t n using the method of iteration. (e) Using your model, when will the temperature of the soup reach 350 F. 3. CSI St. Andrew s. When a murder is committed, the body, usually at 98.6 F, cools according to Newton s Law of Cooling. Suppose that two hours after a murder is committed, the temperature of the body is 95 F, and the temperature of the surrounding air is a constant 68 F.
(a) Let H be a sequence that gives the temperature of the body over time. Write a recursive formula for the sequence H using Newton s Law of Cooling. (b) Find the equilibrium point of this model. How do you know this? (c) Make a spreadsheet that generates and graphs sequence H. Use enough terms to show the long-term behavior. Save your work as time of death SS and print it out. (d) Find an explicit formula for h n using the method of iteration. (e) Suppose a body is found at 4 pm at a temperature of 86 F. Using your model, determine the time the time of death. Newton, Sir Isaac (1642-1727), mathematician and physicist, one of the foremost scientific intellects of all time. Born at Woolsthorpe, near Grantham in Lincolnshire, where he attended school, he entered Cambridge University in 1661; he was elected a Fellow of Trinity College in 1667, and Lucasian Professor of Mathematics in 1669. He remained at the university, lecturing in most years, until 1696. Of these Cambridge years, in which Newton was at the height of his creative power, he singled out 1665-1666 (spent largely in Lincolnshire because of plague in Cambridge) as "the prime of my age for invention". During two to three years of intense mental effort he prepared Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) commonly known as the Principia, although this was not published until 1687. As a firm opponent of the attempt by King James II to make the universities into Catholic institutions, Newton was elected Member of Parliament for the University of Cambridge to the Convention Parliament of 1689, and sat again in 1701-1702. Meanwhile, in 1696 he had moved to London as Warden of the Royal Mint. He became Master of the Mint in 1699, an office he retained to his death. He was elected a Fellow of the Royal Society of London in 1671, and in 1703 he became President, being annually re-elected for the rest of his life. His major work, Opticks, appeared the next year; he was knighted in Cambridge in 1705.
As Newtonian science became increasingly accepted on the Continent, and especially after a general peace was restored in 1714, following the War of the Spanish Succession, Newton became the most highly esteemed natural philosopher in Europe. His last decades were passed in revising his major works, polishing his studies of ancient history, and defending himself against critics, as well as carrying out his official duties. Newton was modest, diffident, and a man of simple tastes. He was angered by criticism or opposition, and harboured resentment; he was harsh towards enemies but generous to friends. In government, and at the Royal Society, he proved an able administrator. He never married and lived modestly, but was buried with great pomp in Westminster Abbey. Newton has been regarded for almost 300 years as the founding exemplar of modern physical science, his achievements in experimental investigation being as innovative as those in mathematical research. With equal, if not greater, energy and originality he also plunged into chemistry, the early history of Western civilization, and theology; among his special studies was an investigation of the form and dimensions, as described in the Bible, of Solomon's Temple in Jerusalem. (The above information was taken from http://www.newton.cam.ac.uk/newtlife.html and Microsoft Encarta.)