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The Case of the Cooling Corpse
6
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 Lastrade. The wealthy but cruel man lied on the floor of his study. It was about 4:30 am, and they did not waste time getting started. The first thing they did was inspect the crime scene and the body. Immediately after, they gathered their suspects and interrogated thier suspects.
The only clues Holmes had to go off of were times and temperatures. He somehow had to figure out who murdered the man with a mere 4 pieces of information. He went directly to the medical examiners office (who also happened to be a friend) for help. He was very good with this kind of stuff.
At the Examiners Office
At the Examiner's Office
"There is a concept we might be able to use," said the medical examiner. "It's called Newton's Law of Cooling. It will essentially help us figure out how much time has passed since he died based on his body temerature." tell me everything you know. Time, temperature, everything."
And so Sherlock told him everything that he and Watson observed...
Observations
"When we arrived at 4:30 am, his temperature was 88°, and two hours later it was 85.8°. The servants said that the room is always kept at76°. The average body temperature is 98.6°, so because he did not seem to be sick around the time of his death, that must have been his temperature when he died. Unfortunately, that was all we could gather at the scene."
The Math
"Don't worry. We can work with that," the examiner replied as he immediatly started scribbling in his notebook.
"What are you doing now," said Holmes, peering into his notebook.
"Well, its really just a simple substitution problem using Newton's law of cooling. Look closer and I can explain it to you."
Newton's Law of Cooling
Newtons Law of Cooling
"We basically know that the rate at which the body's temperature is changing per hour (dt/dy) at a given time (t) is equal to the temperature of the body at that time (y) minus the temperature of the room (r) - which you said was 76° times a constant value (k) that represent the rate at which the temperature of something changes. The only issue is, we are trying to find t, so this equation can't help us as it is. With some derivation using something called integrals, we can come up with an equation that will help us find t using y... y = Ae^tk+r."
More Math
"A little confusing. I know."
"More than a little. Anyway, whats next? We have to find A and K don't we."
"You're catching on quick. Yes, we can use the temperatures you gave me to find them. How far apart did you say you took the temperatures again?"
"Two hours."
"I see... "
Even More Math
"So basically, at t = 0 hours, y = 88°. And at t = 2 hours, y = 85.8°. Using these values, we can find the values of A and k."
After a few minutes...
"Got it! A = 12 and k = -.101. That would make our final equation y = 12e^t(-.101)+78. We want to know t when the man was murdered - so when he y = his normal body temperature. The average resting temperature is 98, so all we have to do is plug that in for y and we get ... t = -6.268 hours.
10:14
"So this means - "
"He was killed 6 hours and 16 minutes before his death. At 10:14 pm." Sherlock quickly answered.
"Precisely."
"I know who did it," said Sherlock.
The Suspects
"He was a cruel man with many enimies, but none had reason enough to murder him. None except his wife, business partner and butler. We listened to their stories and got their alibis, and these are the times each suspect was accounted for:
The wife was accounted for from 10:30 to 11pm.
The business partner was accounted for from 10 to 10:30 pm.
And lastly, the butler was accounted for from 10:05pm till the rest of the night. There is one person here who can provide no information on their whereabouts at 10:14pm, when the murder took place..."
The Answer
"His wife" they both said at once.
“But why? They always seemed like they had a happy marriage?” the examiner said with disbelief.
“To the man it was. But a little digging showed me that his wife never loved him in the first place. Their marriage was just a part of a revenge plot she had.”
“What do you mean?”
“When she was younger, her father ran a multi-billion dollar company - it was world renowned. I’m sure you’re familiar with it.”
“I am. It started out as a joint venture with two founders, but then one left. She sold her shares to the other owner; the wife’s father.”
“That is only partially true. It did start out as a joint venture, but she did not voluntarily leave the company. They had two different visions for the company. Hers was more about philanthropy, and his was strictly about gaining more status and money - no matter the cost. When his partner refused to cut corners to build up their company, he sabatoged her by lying on financial statements. Basically, he accused her of fraud and she was forced to leave the company. In order to avoid unwanted press, they made it look like she willingly left the company.
“It turns out that our victims’s father was the one who screwed our murderer’s father over, and she wanted revenge. She married him with the intent of slowly gaining control of his company over time. She was playing a game that required immense patience and self control, which unfortunately she did not have. As time went on and he got more and more successful, it became harder for her to watch. She couldn’t deal with it anymore and decided to kill him. In all of her rage, she lacked patience and clear thinking. Her plan was poorly excecuted.”
The Other Suspects
“And what about the business partner and the butler?”
“In his climb to become a successful businessman, he cut many corners and hurt lots of people. His business partner had recently discovered information that could end his entire business, and when she confronted him about it, he couteracted and threatened to release information that could completely obliterate her entire business. I can’t speak on it in detail because there is currently an investigation afoot, but just know that she could not risk that information coming out.
“As for the butler, the servants claimed to have heard a screaming match between them coming from the study. Something about the butler’s son.”
“You mean the one who died?”
“Yes. Another mystery for another time. Thank you Doctor for all your help.”
“Don't mention it." He paused. "If anything comes out about their son…”
“You'll be the first to know.”