You and your lab partner(s) will submit one write-up for your group, following the instructions on the back page. Be sure to answer all of the questions asked and follow the directions explicitly. Your final write-up is to be word-processed though the mathematics may be neatly written in by hand in ink.

The body of Magda Huffington, CEO of Important Company, was discovered by her secretary, Dennis Smart, when he reported for work as usual at 8:30 am on Monday. He promptly called 911, and the police were on the scene in minutes. Joe Brown, homicide detective, arrived two minutes after that. The secretary had not seen his boss since Friday. Questioning revealed that no employee admitted seeing the CEO over the weekend. All seemed to have reasons to hate their boss, so there were many suspects.

The office building is located in Arlington, just outside Washington, DC. Joe, the homicide detective, made a list of some of the people who had access to the building between Friday afternoon and Monday morning:

1. a) The janitorial crew, Emilio and Thorne, work an 11 pm to 7 am shift each night, Sunday through Thursday.

2. b) The CEO's ex-husband, Rocky Atlas, still has a key to the building. He could not account for his whereabouts over the weekend, but he had reported at 9 am Monday to his regular job in Cleveland, Ohio.

3. c) Bubba Huffington, the CEO's weak-willed nephew and heir to her fortune, played tennis most of Saturday and his alibi was confirmed by his tennis partner. He claims that he went sailing alone most of Sunday, and his wife asserts that he was in bed with her from midnight Sunday to about 9 am Monday when he was notified of his aunt's death.

4. d) The vice-president, Mark Clark, is the real brains and the chief operating office of the company. He had been passed over when the appointment of CEO was made. He is known to work from 6:30 am to 6:30 pm every day but Sunday. On Sunday, Mark attended two church services and visited the sick in two hospitals. He had been at work two hours when Dennis made his grisly discovery, but he always made it a practice to avoid meeting Ms Huffington except for board meetings.

Since the body of the murder victim has been cooling since the time of death, Detective Joe knows that the temperature of the body can indicate when Ms Huffington died. When he arrives on the scene, he ascertains that the room temperature is 21 Celsius. Detective Joe knows some calculus so he knows Newton's Law of Cooling, which tells him that the rate of change of the body temperature is

* Problem taken from Calculus: A Modeling Approach by David Smith and Lawrence Moore

proportional to the difference between the body temperature and the room temperature, that is

dB dt =k(B21)

where B is the temperature of the cooling body in degrees Celsius at time t in hours. To find the exact time of death, Joe knows he needs an initial value condition. He notes that at 8:50 am, the temperature of the body was 30 Celsius, so his initial value problem is

dB dt =k(B21) where B(0)=30

You must help Joe solve this problem. Once you and Joe know the time of death, together you can, perhaps, discover the culprit.

1. 1) You have a differential equation with a negative growth factor (that is, the k). Make some comments on the relevancy of the growth factor being negative, that is,

   discuss why you should expect that the derivative dB is negative. dt

   dB

2. 2) Using any of our methods, solve the differential equation dt = k(B 21) for B .

   dB

3. 3) By substituting your B and its derivative into the differential equation dt = k(B 21) ,

   show that your B actually is a solution to the differential equation.

4. 4) You still don't know the value of k , and for that Joe knows that he must take another body temperature. At 9:50 am, the body temperature has dropped to 28 Celsius; that is, B = 28 when t = 1 hour. From this information, can you calculate k ? Do it!

5. 5) Joe believes that, at the time of death, the body temperature was 37 Celsius. Why can he make this assumption? Why 37 Celsius??

6. 6) Find the time t when B = 37 . Why is this value of t negative? Exactly what time of day does this value represent?

7. 7) Now that you know the time of death, help Joe sort out the suspects and write his report to his supervisor, Captain Nancarrow. Are there any additional suspects? (You may create as many as you wish.) Locate the guilty party and determine a motive.