OSH/6

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A dead body was found at 8 am on Nov 13 in the Rose Garden on campus. The victim was last seen alive at 10 pm the night before. The temperature of the body was measured at 8 am and found to be 22 C. Three hours later, the body’s temperature was measured again, just before being moved to the morgue, and was found to be 19.5 C. The temperature in the area was tracked at a nearby weather station and found to match well to the function

\(\displaystyle E(t)=12-3\cos\left(\frac{2\pi t}{24}\right)\)

where \(t\) is clock time measured in hours.

Assume that Newton’s Law of Cooling holds:

\(\displaystyle \frac{dT}{dt} = k (E(t)-T(t)) \)

Note that the substitution method discussed in class for solving this equation does not apply here because \(E(t)\) changes with time.

  1. What is the period of the function \(E(t)\)? When does it reach a maximum? minimum? What is the amplitude?
  2. Use a spreadsheet to determine what value of \(k\) gives an approximate solution \(T(t)\) satisfying both \(T(8)=22\) and \(T(11)=19.5\). Use a step size of \(\Delta t=1/4\) hr. Hint: Think of \(T(8)\) as an initial condition and create a spreadsheet to solve the differential equation with some randomly chosen value of \(k\) and then change strategically \(k\) until the condition at \(t=11\) is satisfied. Do not hand in your spreadsheet. Instead, describe in words how you set up the spreadsheet to solve the problem giving formulas used in any of the important columns used.
  3. Using the value of \(k\) found in #2, determine the time of death (to the nearest 1/4 hr) assuming the body was 37 C when the murder occurred. Hint: you will have to solve the differential equation backward in time from \(t=8\) for an unknown period of time (but no farther back than 10 pm). Do not hand in your spreadsheet. Instead, describe in words how you set up the spreadsheet to solve the problem.