LOG PROJECT: MURDER MYSTERY

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1 Extra Credit Due Date: 05/11/2015 LOG PROJECT: MURDER MYSTERY You ve all been shipwrecked on a tropical island - a wonderful place, with bananas, coconuts, fish, and a pretty constant temperature of 80. Your teacher, Mr. Roemer, has been murdered. You know there s no one else on the island - it was one of your classmates that did it! None of you will sleep peacefully in your flimsy grass huts until the murderer is discovered. You have watches, thermometers, and other simple tools, but no experts on murder investigations. The day Mr. Roemer was murdered, everyone was walking around the island by two s, noting its features, in hopes that would help you all figure out where you are. It turns out that everyone walked by the spot where Mr. Roemer s body was found, and recorded the time when they were at a nearby spot (where they could see a volcano on the next island). Figuring out the time of death would likely narrow down the suspects to four or less. Nic had a hunch that knowing the body temperature would help determine the time of death. So, at 1 pm, he checked the temperature of Mr. Roemer s corpse. It was Then at 2 pm, it was Finding the murderer will be our goal. It might take us a few days. (I hope you don t mind some sleepless nights...) One more clue this comes from Mrs. Boling, I read a lot of murder mysteries. In one of them, this detective says, A dead body cools off just like a hot cup of joe. I don t know if that helps or not You ll also need to know: = + Where: B: is the body temp (usually 98.6 ) A: is the temperature of the air E: is how much hotter the body was than the air when first measured. b: is the ratio of the temperature t: is the time measured in hours.

2 PART 1 We want to think about how a hot cup of coffee cools off. y 1. What would be a reasonable starting temperature? 2. After about how long would it be cold? 3. About what temperature is it when it s cold? (Why?) 4. Now let time be the x-axis (t-axis) and temperature (T) be the y-axis, and graph temperature versus time for a cup of coffee, using what you know from common sense. Does a straight line graph make sense for this? x PART 2 Sara says We need some numbers here. And she boils some coffee up over a campfire and measures its temperature with a thermometer Dylan provides. Here s what she gets: y The coffee starts out at 176 degrees, and cools off like this 160 Min Degrees x 5. If he measured the coffee at 2 hours and 3 hours, what temperature would it be? 6. Graph this data, and connect the points with a smooth curve. 7. So we can conclude that this graph has what line as an asymptote? 8. Give an example of a function with this asymptote.

3 P ART 3 Mr. Roemer s dead body was lying near the viewing spot for the volcano, and it turns out that there was only one path going by that spot. So, after checking with each other, and remembering who passed whom, you all agree that the murderer was most likely one of the people at that spot right before or after the time of death. Below are the times that each pair walked by the volcano viewing spot: Sara and Ana Baylee and Nic James and Carlos Dylan and Natalie Kaleb and Cynthia Jacob and Robert Seth and Markus Leah and Logan Joshua and Jamison 12. 9:05 9:20 9:35 9:50 10:05 10:20 10:35 10:40 10:55 When you figure out the time of death, you ll know the 4 most likely suspects. 10. A: 11. Initial body temp: E: (How much hotter was the initial body temp than the air temp?): 13. b (Ratio of current temperature over initial temperature taken): 14. Write the function: ( ܣ = ܤ + ܧ ௧ ): 15. Rewrite the exponential function as a log: 16. Solve for t: 17. Time of Death (absolute value of t hours BEFORE initial time): 18. Suspects: 9. B:

4 P ART 4: T HE N EXT D AY Of course, all 4 suspects swear they re innocent. The next day, Baylee finds Mrs. Boling dead, left lying right in the clearing. My body temperature is 93.1, and it s 9:46pm. You check at 10:16pm, and it s Here s everyone s alibis: James, Jamison, Seth, and Joshua were all swimming together from 8am to 9:30am. Sara, Natalie, Dylan, Ana, and Carlos were all looking for clams together from 8:30am to 9:30am. Robert, Jacob, Nick, and Baylee were all gathering coconuts together from 9am until they heard Mrs. Boling s screams. Kaleb, Logan, Leah, Cynthia, and Markus were hiking from 9:30 until they heard the screams. Please find the killer before someone else is murdered! 19. B: 20. A: 21. Initial body temp: 22. E: (How much hotter was the initial body temp than the air temp?): 23. Time between temperatures taken (in hours): 24. b (Ratio of current temperature over initial temperature, and then raised to the power of the time between temperatures): 25. Write the function: ( ܣ = ܤ + ܧ ௧ ): 26. Rewrite the exponential function as a log: 27. Solve for t: 28. Time of Death (absolute value of t hours BEFORE initial time):

5 29. Suspects: 30. Now you re the prosecuting attorney, and you need to write up a closing argument explaining to the jury how we know the time of death, and why that means the prime suspect is the killer.