Case for Murder

Question

The evening cleaning crew at a large tech firm discovers a dead body in the early morning hours. Police are immediately dispatched to the scene. Upon arrival at 5:00 am, police measure the temperature of the body to be 88.2°F. An hour later, the coroner arrives and again measures the temperature of the body. This time, the temperature is 86.7°F. The environmental system of the building maintains a constant temperature of 67°F. Complete the following, including detailed explanations of your work.

1. Determine a formula of the form y = a(b) x + c that models the temperature of the body as a function of the time (in hours since the police arrived). Round any values to two decimal places.

2. Graph your function on a reasonable domain. Label and scale your graph.

3. Discuss what will happen to the body temperature if it were never discovered (or at least not for many days), and explain what this represents mathematically.

4. Use your function to estimate the time of death, to the nearest half-hour. Provide an algebraic solution and include a labeled Desmos graph illustrating your solution. Assume normal body temperature is 98.6°F.

Answer 1

I would model this as dT/dt = k(T-67), i.e. T(t) = 67 + Ae^kt

I would assume that k is constant from time of death until at least 1 hr after police discovery.

Start at 5PM 88.2=67+ A to get a value for A

Then 1 hr later 86.7 = 67 + Ae^k; this gives you a value for k.

Then use this k with 88.2 = 67 + 31.6e^kt to estimate the time of death.