Question

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 subscript e plus left parenthesis T subscript 0 minus T subscript 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 subscript e over denominator T subscript 0 space end subscript minus T subscript 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).

Answer 1

Solving an exponential equation, we can assume that the time of death is 11:42.6 am

How to get the time of death?

The information we care here is:

• The initial temperature of the person is assumed to be T₀ = 98.6 °F
• The temperature of the room is  Tₐ = 70°F
• The temperature (of the body) at the time t is 78 °F
• The constant is K = 0.1335

Now, the Newton cooling equation is:

T = (T₀ - Tₐ)*e^{-k*t} + Tₐ

Replacing the values that we know, we get:

78°F = (98.6°F - 70°F)*e^{-0.1335*t} + 70°F

Now we can solve that equation for t.

78°F = (98.6°F - 70°F)*e^{-0.1335*t} + 70°F

78°F - 70°F = (98.6°F - 70°F)*e^{-0.1335*t}

8°F = (28.6°F)*e^{-0.1335*t}

8/28.6 = e^{-0.1335*t}

If we apply the natural logarithm in both sides:

ln(8/28.6) = ln( e^{-0.1335*t} ) = -0.1335*t

-ln(8/28.6)/0.1335 = t = 9.54

So the person was dead 9.54 hours before 9:15pm

Where 9.54 hours = 9 hours + 0.54 hours

Let's subtract the minutes first:

And we know:

1 hour = 60 min

0.54 hours = 0.54*60 min = 32.4 min

9:15 - 32.4 mins = 8:42.6

Now we can subtract the hours:

8:42.6 - 9 hours = 11:42.6 am

So we can assume that the time of death is 11:42.6 am

If you want to learn more about exponential equations:

https://brainly.com/question/11832081

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