help out with this word problem

Question

The Facts:

A police man has just been contacted about a possible murder in California. The lifeless body of a human, Mr.X was discovered at 7:23 am on the morning of January 22 nd in his home office. It was also noted that his body temperature was 84.1°F when discovered, and by 8:42 am his body temperature was 83.5°F. Forensic agent also recorded the temperature in the room to be 73.1°F.

The suspect were 3 and were immediately identified upon the initial investigation. Mr.S, another person who works with him has been said to have always been jealous of his co-worker Mr. X, because mr x stole Mr.s girlfriend Mr.s. has an airtight alibi for portions of yesterday, as he was in his office from 7:00 am to 11:30 am, then took a lunch break and was back in his office from 1:00 pm to 5:00 pm.

Another suspect of interest is a local small time crook Mr. J . It is well known that he always held a grudge against Mr. X for his refusal to lend him money. Mr J also has an alibi for most of the previous day, as from 8:30 am to 2:00 pm he was helping keep order in the operations of his business. He then spent the late afternoon and evening with his girlfriend, Ms. P, from 4:30 pm to 6:30 pm.

The last suspect in the death of Mr. X was Ms.Y. Ms. Y held a grudge against Mr. X ever since she asked him for a favour but he never did it, Although this was not the first time Mr. X did not do her favour it was definitely the last. MS.Y alibi holds strong from 11:30 am to 4:30 pm as she was working her second job at McDonald's

The Task:

forensic has prepared you with the mathematical skills you need to solve the problem. Recall the following about the rate at which objects cool:

Newton’s Law of Cooling

The rate of change in the temperature of an object is directly proportional to the difference between the object’s initial temperature 𝑇0 and the temperature of the surrounding medium 𝑇𝑀 in which it is placed. The temperature 𝑇(𝑡) of the object at time 𝑡 can be modeled by

𝑇(𝑡) = 𝑇𝑀 + (𝑇0 − 𝑇𝑀)𝑒𝑘𝑡

It is now up to you to solve the crime. Determine the exact time of death for Mr. X AND then use it to determine which of the three suspects was his murderer. You may assume that at the time of death, Mr. X's body temperature was 98.6℉.

Answer 1

The steps for solving the problem:

1) Use the two body temperatures and times you know (84.1 at 7:23, 83.5 at 8:42) to figure out the k in the equation T(t) = T_M + (T₀ - T_M) e^kt

2) Then use the final temperature and time (83.5 at 8:42) and the starting temperature (98.6), along with the k, to figure out the time t in the equation.

For the first step, we set the final temp T(t) = 83.5, the starting temp T₀ = 84.1 and the room temp T_M = 73.1. And we need to figure out t, the time between the the starting and final temps. A good way is to convert into minutes from midnight, so 7:23 = 7 x 60 +23 = 443 and 8:42 = 8 x 60 + 42 = 522. So t = 522 - 443 = 79 minutes.

So, the equation is 83.5 = 73.1 + (84.1 - 73.1) e^79k

Which is (83.5 - 73.1) / (84.1 - 73.1) = e^79k

Now take the logarithm of both sides: ln ((83.5 - 73.1) / (84.1 - 73.1)) = ln (e^79k) = 79k

Divide both sides by 79 and get: ln ((83.5 - 73.1) / (84.1 - 73.1)) / 79 = k

Plug this into the calculator and get k = -.00071 rounded.

For the second step, we use a different starting temp T₀ = 98.6, Mr. X's temperature when he was alive.

So, the equation is 83.5 = 73.1 + (98.6 - 73.1) e^{-.00071 t}

And we can solve for t using the same techniques shown above, and get t = 1263 minutes rounded. That's how long Mr. X has not been alive.

So Mr. X was killed 1263 minutes before 8:42 am. This goes into the previous day. So a good way to convert 8:42 am into the number of minutes from the previous midnight. For 8:42 is 32 hours and 42 minutes from previous midnight, which gives 32 x 60 + 42 = 1962 minutes.

So Mr. X was killed at 1962 - 1263 = 699 minutes after the previous midnight. To covert that into hours:minutes, we need to divide by 60 for the whole number hours and use the remainder for minutes, which gives 11:39 am the previous day.

Using that time, you can look at the alibis of the three suspects and figure out which one could have done the crime.