I recommend reading Dunham’s book as a source of inspiration for writing lesson plans. This lesson plan is an effort to incorporate more real-life math into the curriculum and find topics that might interest the student, show them a reason to study math and give them an opportunity to use available technology. This lesson explores the math used in forensic science, looks at Newton’s Law of Cooling, reviews exponential functions, requires modeling techniques, and makes good use of graphing calculators. The demonstration involves a cooling baked potato (Mr. potato head) acting as the dead body, and a detective (student) who is trying to determine the time of death. Objective: Students will see how exponential functions can be used by a detective. Students will gain practice in accumulating data, modeling and using functions. Students are then given a bit more information about Euler’s number ‘e’ and exponential functions so that they can improve their best fit equation. A review of natural logarithms may be necessary. -------------------------------------------------------------------------------------------------------------------------- Student Worksheet
Scenario: Mr. Potato Head was found dead in his apartment around midnight. He died sometime during the night. When a ‘person’ dies, their body temperature begins to cool. The student (detective) needs to determine the time of death. Procedure: Accumulate data using a Celsius thermometer and a watch, or a CBL unit and temperature probe and graphing calculator. Start by noting the air temperature in the classroom and the ‘body’ temperature at approximately 5 minute intervals as the potato cools. A table of values should be recorded showing body(potato) temperature and time converted to hours. When these values are graphed, begin determining the equation of the graph. Graph your data and attempt to derive an equation that best fits the data and be ready to justify your equation. Using the trace or table features of your graphing calculator, estimate the time of death. Your best equation: Estimated time of death : Although this equation may not be the best equation this is an important part of modeling and can give a good estimate of time of death. Newton’s Law of Cooling states that the rate at which an object cools is proportional to the difference between the object’s temperature and the temperature of its surroundings. In other words when there is a big difference between the temperatures of the surrounding air and the body, the body cools rapidly. A small difference in the two temperatures causes a slower rate of cooling.
Temperature of the body at a certain time, T(0) = initial air temperature + (Difference in body and air temperatures at T(0) divided by e^(rt) )
Using function notation you could write T(t) = A(0) + (T(0) - A(0))/ exp(rt) where r is a rate of cooling number (0<x<1) and t is the time in hours. T(0) is the temperature of the body and A(0) is the air timperature in the apartment at t=0 (time found). Find the constant r by plugging in a few data points from your table, or by xperimenting with different values for r until the graph passes through most of the data points from the table. This kind of trial and error experimentation is an important and creative side of mathematical modeling.
Ask the teacher for the normal living body temperature fo Mr. Potato Head (He/She will give you the temperature noted right out of the microwave) Note: that if we were using a real body we would be using 98.6 degree F. Calculate the time of death. A review of using natural logarithms may be necessary here. For example if the air temperature was 50 degrees C at the time the body was discovered and the temperature of the body at time found was 56 degrees C, then the equation might look like this:
T(t) = 50 degrees C + 6 degrees C / exp(.5t)
To find the time of death get exp(.5t) on one side of the equation and take the log of both sides
exp(.5t) = 6/(T(t) - 50)
ln (exp(.5t)) = ln( 6/(T(t)-50))
Plug in the normal body temperature T(t) and use your calculator to solve the right side of this equation. Let's say that the teacher noted the temperature out of the microwave to be 59 degrees (time of death).
ln (exp(.5t)) = ln (6/(59-50))
.5t = -.40547
t = -.81093
Congratulations! The time of death would be .81093 hours before the body was discovered. If the body was discovered at midnight, then the time of death was around 11:11 p.m. Since your experiment uses a different temperature out of the microwave, what did your investigation find?? _______ p.m.
Now try this one: At midnight, police were called to the scene of a brutal murder and found the body of Neils Nieley. The officer immediately noted that the air temperature in the apartment was 68 degrees F and Niels' body temperature was 85 degrees. After the police were finished with their search and fingerprinting at 2:00 they checked the body temperature once more and found it had cooled to 75 degrees. The police arrested Niels' wife Narley Nieley and charged here with the murder. She had an eyewitnesses that said she left Ned's Bar at 11:15 p.m. She had just been jilted by Neils and was a good suspect.
Narley's lawyer Nelson Noodle knew about Newton's law of cooling and used the function
T(t) = 68 degrees + 17 degrees/ exp(.5207 t)
to find the time of death. Can you help him prove that Narley could
not have done it? Explain and show your calculations below. Are there ary
other factors of cooling bodies that should be considered in order to make
this process even more accurate??