A Kern High School District Task

Shipwreck and Murder

You’ve all been shipwrecked on a tropical island - a wonderful place, with bananas, coconuts, fish, and a pretty constant daytime temperature of 75°and a constant nighttime temperature of 65°.

Your classmate, John Doe, 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 John was murdered, everyone was walking around the island 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 John'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 will likely narrow down the suspects.

MaryAnn had a hunch that knowing the body temperature (normal body temperature is 98.6°) would help determine the time of death. So, at 1 pm, she checked the temperature of John’s corpse. It was 91.3°. Then at 1:40 pm, it was 87.5°.

One more clue… this comes from Daniel, “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…”

We want to think about how a hot cup of coffee cools off.

1. What would be a reasonable starting temperature?

We want to think about how a hot cup of coffee cools off.

2. About what temperature is it when it’s cold? (Why?)

We want to think about how a hot cup of coffee cools off.

3. How long would it take to reach the “cold” temperature?

Let’s Investigate

The following data represents the temperature of a cup of coffee as it cools over time.

Time Temperature

10:00pm 15410:02pm 14210:03pm 13810:05pm 13110:10pm 11010:15pm 9710:20pm 8510:30pm 7511:00pm 66

Graph the data on the Shipwreck and Murder worksheet.

Based on the graph of the data, do you think the coffee will ever reach a temperature of 60? Explain your answer.

What is the lowest temperature the coffee will reach? How did you determine this?

What type of function could be used to model the data?

What would be the general form of the equation for this type of function?

Which unknown value in the equation would be the easiest to find first? How do we find that value?

To find the values for and choose any two points from the data set and plug them into the equation. You will now have a system of equations.

Use your equation to approximate the temperature of the coffee after 25 minutes. Does your answer seem reasonable based on your graph and table?