Example: Newton’s Law of Cooling states that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature and the temperature in the surrounding medium. A detective finds a murder victim at 9 am. The temperature of the body is measured at 90.3 °F. One hour later, the temperature of the body is 89.0 °F. The temperature of the room has been maintained at a constant 68 °F.

(a) Assuming the temperature, T, of the body obeys Newton’s Law of Cooling, write a differential equation for T.

(b) Solve the differential equation to estimate the time the murder occurred

Newton’s Law of Cooling

One common method of shaping plastic products is to pour hot plastic resin into a mold. The resin, which was poured at a temperature of 300 F, is then cooled in a chiller system that is kept at 58 F. After cooling to an appropriate temperature, the molded product is ejected. This is done as quickly as possible to minimize cost and speed up production. Ejecting the product too quickly, however, can introduce warping or punctures.

Apply the differential equation and its general solution from the example to the cooling plastic. Is there enough information to determine the value of the proportionalityconstant k? If so, find its value. If not, explain what additional information isneeded to determine the value of k.

A cup of coffee is initially 170 degrees Fahrenheit and is left in a room with ambient temperature 70 degrees Fahrenheit. Suppose that when the coffee is first placed in the room, it is cooling at a rate of 20 degrees per minute. Assuming Newton's law of cooling applies, how long does it take for the coffee to cool to 110 degrees?

The Case of the Cooling CorpseIt was a dark and stormy night.  Holmes and Watson were called to the scene of the murder by Inspector Lestrade of the police.  The victim was a wealthy but cruel man.  He had many enemies.

The most likely suspects are the wife, the business partner, and the butler.  Each has an equally strong motive.  Each also has an alibi.  The wife claims to have spent the entire evening at the theater across town.  She was seen leaving the theater at 10:30 p.m. and returned home at 11:00 p.m., going straight up to her bedroom.  Her return was verified by the upstairs maid.  The business partner claims to have spent the evening working on papers at the office.  His wife and household staff verified that he returned home at 10:30 p.m.  The butler was on his night off.  He claims to have been at the local pub until 10:00 p.m.  The butler returned to his quarters above the carriage house at 10:05 p.m. and did not leave.  This was verified by the other servants.

The body was found in the victim’s study.  Holmes arrived at the scene at 4:30 a.m.  The room was unusually warm and stuffy.  One of the police officers went to open a window.  Holmes admonished him to delay that action until he had completed his investigation of the crime scene.  He instructed Watson to determine the temperature of the body.  This was found to be 88.0°F.  Holmes questioned the servants as to the room temperature during the evening and learned that the man had liked the room warm and that the temperature in the study was always very near the current 76°F.  Holmes asked Watson to take the temperature of the body again at the conclusion of his inspection of the scene, two hours after the first reading.  It was 85.8°F.

Your task is to solve the mystery and write the end of this story

The coroner’s office is kept at a cool 17°C.Agent 008 kept pacing back and forth trying to keep warm as he waited for any newinformation about his latest case. For over three hours now, Dr. Dedman had beenperforming an autopsy on the Sideroad Slasher’s latest victim, and Agent 008 could see thatthe temperature of the room and the deafening silence were beginning to irritate even Dr.Dedman. The Slasher had been creating more work than Dr. Dedman cared to investigate.“Dr. Dedman, don’t you need to take a break?” Agent 008 queried. “You’ve beenexamining this dead body for hours! Even if there were any clues, you probably wouldn’tsee them at this point.”“I don't know.” Dr. Dedman replied. “I just have this feeling something is not quite right.Somehow the Slasher slipped up with this one and left a clue. We just have to find it.”“Well, I have to check in with HQ,” 008 stated. “Do you mind if I step out for a couple ofhours?”“No, that’s fine,” Dr. Dedman responded. “Maybe I’ll have something by the time youreturn.”Sure, 008 thought to himself. Someone always wants to be the hero and solve everythinghimself. The doctor just does not realize how big this case really is. The Slasher has left atrail of dead bodies through five states! 008 left, closing the door quietly. As he walkeddown the hall, he could hear the doctor’s voice fade away, as he described the victim’sgruesome appearance into the tape recorder.The hallway from the coroner’s office to the elevator was long and dark. This was theonly way to Dr. Dedman’s office. Didn’t this frighten most people? Well, it didn’t seemto bother old Ajax Boraxo who was busy mopping the floor, thought 008 . . . most otherswouldn’t notice, he reminded himself.He stopped to use the restroom and bumped into one of the deputy coroners. “Dedmanstill at it?” “Sure is, Dr. Quincy. He’s totally obsessed. He’s certain there is a clue.”As usual, leaving the courthouse, 008 had to sign out.“How’s it going down there, Agent 008?” Sergeant Foust asked. Foust spent most of hisshifts monitoring the front door, forcing all visitors to sign in, while he recorded the timenext to the signature. Agent 008 wondered if Foust longed for a more exciting aspect oflaw enforcement. He thought if he were doing Foust’s job he would get a little stir-crazysitting behind a desk most of the day. Why would someone become a cop to do this?“Dr. Dedman is convinced he will find something soon. We’ll see!” Agent 008 responded.He noticed the time: Ten minutes before 2:00. Would he make it to HQ before the chiefleft?“Well, good luck!” Foust shouted as 008 headed out the door.Agent 008 sighed deeply when he returned to the courthouse. Foust gave his usualgreeting: “Would the secret guest please sign in?” he would say, handing a pen to 008 ashe walked through the door. Sign in again, he thought to himself. Annoying! 5:05 p.m.Agent 008 had not planned to be gone so long, but he had been caught up in what the staffat HQ had discovered about that calculator he had found. For a moment he saw a positivepoint to having anyone who came in or out of the courthouse sign in: he knew by quicklyscanning the list that Dr. Dedman had not left.As he approached the coroner’s office, he had a strange feeling that something was wrong.He could not hear or see Dr. Dedman. When he opened the door, the sight inside stoppedhim in his tracks. Evidently, Dr. Dedman was now the newest victim of the Slasher. Butwait! The other body, the one the doctor had been working on, was gone! Immediately,the security desk with its annoying sign-in sheet came to mind. Yes, there were lots ofnames on that list, but if he could determine the time of Dr. Dedman’s death, he might beable to scan the roster to find the murderer! Quickly, he grabbed the thermometer tomeasure the Doctor’s body temperature. He turned around and hit the security buzzer.The bells were deafening. He knew the building would be sealed off instantly andsecurity would be there within seconds.“My God!” Foust cried as he rushed in. “What a travesty! I was talking with Dr.Dedman less than an hour ago.”As the security officers crowded into the room, Agent 008 explained what he knew. Hehad stopped long enough to check the doctor’s body temperature: 27°C. That was 10°Cbelow normal. He figured that the doctor had been dead at least an hour. Then heremembered: the tape recorder! Dr. Dedman had been taping his observations; that wasstandard procedure. They began looking everywhere. The Slasher must have realizedthat the doctor had been taping and taken the tape recorder as well. Exactly an hour hadpassed during the search and Agent 008 noticed that the thermometer still remained inDr. Dedman’s side. The thermometer clearly read 24°C. Agent 008 knew he could nowdetermine the time of death precisely. When did Dr. Dedman die? Who is the murderer?Coroner’s Office - Please Sign InName Time In Time OutLt. Borman 12:08 2:47Alice Bingham 12:22 1:38Chuck Miranda 12:30 2:45Harold Ford 12:51 1:25Ajax Boraxo 1:00 2:30D. C. Quincy 1:10 2:45Agent 008 1:30 1:50Ronda Ripley 1:43 2:10Jeff Dangerfield 2:08 2:48Stacy Simmons 2:14 2:51Brock Ortiz 2:20 2:43Pierce Bronson 3:48 4:18Max Sharp 3:52 5:00Maren Ezaki 3:57 4:45Caroline Cress 4:08 4:23Milly Osborne 4:17 4:39D.C. Quincy 4:26 4:50Vinney Gumbatz 4:35Cory Delphene 4:48 4:57Max Crutchfield 5:04Agent 008 5:05Security 5:12Problem Solving, Reasoning, and Logarithmscreated by: Judith KyshSan Francisco State Universityjkysh@sfsu.eduMaterials adapted from CPM Algebra II Connections (2007)More information at www.cpm.org