A detective discovers a murder victim in a room at the Marriott Marquis Hotel at 9:15 pm on Friday night. Immediately, the temperature of the body is recorded as being 78 °F. The programmable thermostat has been set to 70 °F for the last week. What was the time of death?
Newtons Law of Cooling is an exponential equation, which describes the cooling of a warmer object to the cooler temperature of the environment. Specifically, we write this law as,
T left parenthesis t right parenthesis equals T subscriipt e plus left parenthesis T subscriipt 0 minus T subscriipt e right parenthesis e to the power of negative k t end exponent
where T (t) is the temperature of the object at time t, Te is the constant temperature of the environment, T0 is the initial temperature of the object, and k is a constant that depends on the material properties of the object. To solve this exponential equation for t, you will need to use logarithms. This equation can be rearranged to:
fraction numerator T space left parenthesis t right parenthesis space minus T subscriipt e over denominator T subscriipt 0 space end subscriipt minus T subscriipt e end fraction equals e to the power of negative k t end exponent
Tip: To organize our thinking about this problem, lets be explicit about what we are trying to solve for. We would like to know the time at which a person died. In particular, the investigator arrived on the scene at 9:15 pm, which is t hours after death. The temperature of the body was found to be 78 °F. Assume k = 0.1335 and the victim’s body temperature was normal (98.6 °F) prior to death. Show all work (upload a picture or type in how you solved the problem).
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