Question

Can someone please help me with this problem for my Calculus class? I am not sure how to solve it and am very confused.

Newton’s Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and its surrounding medium. Newton’s Law of Cooling is represented by the formula T=T_m+(T_0-T_m ) e^(-kt) where
T= temperature of the object object after t hours
T_m= temperature of the surrounding medium
T_0=initial temperature of the object
k=rate of change in the temperature of the object
t=time, in hours, that the object has been exposed to the surrounding medium.
One application of Newton’s law is to forensic medicine. A murder has occurred and the facts of the case are these. There are 5 suspects and each has an airtight alibi, except for the following times each was alone during the day.
Miss Scarlet: “home alone” from 10:30-11:45am
Professor Plum: “stuck in traffic” from 11:45am-1:30pm
Mrs. Peacock: “baking cookies” from 1:30-3:45pm
Mr. Green: “gone fishing” from 3:45-5:00pm
Col. Mustard: “can’t recall” from 5:00-6:15pm
The butler has no alibi. So if the murder did not occur between 10:30am and 6:15pm, the butler must have done it!

The coroner arrived at the scene at 9:00pm and determined the temperature of the corpse to be 82.7°F. The room temperature is kept at exactly 65°F. One hour later, the body temperature has dropped to 80.8°F. Assuming that the victim was healthy ( before the murder, of course) and had a normal temperature of 98.6°F, who is the killer and when did the murder occur? Be precise to the second.

Answer 1

Mr. Green is the murderer.

Explanation:

Forget all the word problem bs. Just focus on the numbers.

You're given the equation

`T=T_m + (T_0-T_m)e^(kt)`

`T_0=` initial temp

`T_m=` temp of room

`T =` temp after t hours

`k=` how fast the temp is changing

t = time (hours)

1.) We're given:

It took 1 hour for the body to go from 82.7° to 80.8° when the police found the body. The room temperature was 65°

`T_0=` 82.7°

`T_m=` 65°

`T =` 80.8°

t = 1

`k=` ???

the only thing we don't know is the value for k, so lets solve for it.

REFER TO STEP (1) in the picture.

You should get that
`k=ln((15.8)/(17.7))` . This is how fast the body was loosing heat.

2.) Now we know what k is (how fast the temperature is dropping), we can use it to find how long it took for his fresh body (98.6°) to drop to 82.7° when the police found it.

`T_0=` 98.6°

`T_m=` 65°

`T =` 82.7°

t = ???

`k=ln((15.8)/(17.7))`

We don't know what t is so lets solve for it. t will tell us how long before the police arrive (9:00pm) did the murder happened.

REFER TO STEP (2) in the picture.

You should get t=5.5079 hours. This means the murder happens 5.5 hours before 9:00pm.

3.) Since you know that the murder happened 5.5 hours before 9:00pm, subtract it.

9:00 - 5.5hours = 3:30pm

The murder happened around 3:30pm.

Mr. Green is lying because he said that he was fishing around 3:45pm-5:00pm, but that's when the murder occured.