Newton’s Law of Cooling
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
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?
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 proportionality
constant k? If so, find its value. If not, explain what additional information is
needed to determine the value of k.
The Case of the Cooling Corpse
It 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 new
information about his latest case. For over three hours now, Dr. Dedman had been
performing an autopsy on the Sideroad Slasher’s latest victim, and Agent 008 could see that
the 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 been
examining this dead body for hours! Even if there were any clues, you probably wouldn’t
see 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 of
hours?”
“No, that’s fine,” Dr. Dedman responded. “Maybe I’ll have something by the time you
return.”
Sure, 008 thought to himself. Someone always wants to be the hero and solve everything
himself. The doctor just does not realize how big this case really is. The Slasher has left a
trail of dead bodies through five states! 008 left, closing the door quietly. As he walked
down the hall, he could hear the doctor’s voice fade away, as he described the victim’s
gruesome appearance into the tape recorder.
The hallway from the coroner’s office to the elevator was long and dark. This was the
only way to Dr. Dedman’s office. Didn’t this frighten most people? Well, it didn’t seem
to bother old Ajax Boraxo who was busy mopping the floor, thought 008 . . . most others
wouldn’t notice, he reminded himself.
He stopped to use the restroom and bumped into one of the deputy coroners. “Dedman
still 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 his
shifts monitoring the front door, forcing all visitors to sign in, while he recorded the time
next to the signature. Agent 008 wondered if Foust longed for a more exciting aspect of
law enforcement. He thought if he were doing Foust’s job he would get a little stir-crazy
sitting 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 chief
left?
“Well, good luck!” Foust shouted as 008 headed out the door.
Agent 008 sighed deeply when he returned to the courthouse. Foust gave his usual
greeting: “Would the secret guest please sign in?” he would say, handing a pen to 008 as
he 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 staff
at HQ had discovered about that calculator he had found. For a moment he saw a positive
point to having anyone who came in or out of the courthouse sign in: he knew by quickly
scanning 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 stopped
him in his tracks. Evidently, Dr. Dedman was now the newest victim of the Slasher. But
wait! 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 of