If you take a hot dinner out of the oven and place it on the kitchen countertop,

Question

If you take a hot dinner out of the oven and place it on the kitchen countertop, the dinner cools until it reaches the temperature of the kitchen. Likewise, a glass of ice set on a table in a room eventually melts into a glass of water at that room temperature. The rate at which the hot dinner cools or the ice in the glass melts at any given time is proportional to the difference between its temperature and the temperature of its surroundings (in this case, the room). This is called Newton’s law of cooling (or warming) and is modeled by

Explanation / Answer

T(t) = Ta + (Th - Ta)e^(-kt)

where,

T(t) = Temperature at time t,
TA = Ambient temperature (temp of surroundings),
TH = Temperature of hot object at time 0,
k = positive constant and
t = time.

We are given Ta = 60...

T = 60 + (Th - 60)e^(-kt)

Let t = 0 be 4 am...
So, when t = 0, T = 87 :
87 = 60 + (Th - 60)e^0
Th = 87

So, we have
T = 60 + 27e^(-kt)

Now, when t = 1, i.e 5 am. T = 78....
78 = 60 + 27e^(-k)
e^-k = 2/3
k = -ln(2/3)
k = 0.405465

So, we have :

T = 60 + 27e^(-0.405465t)

Now, when was the person dead?
Thisll happen when T = 98.6....

98.6 = 60 + 27e^(-0.405465t)

e^(-0.405465t) = 1.4296296296296296

-0.405465t = ln(1.4296296296296296)

t = -0.8814950993709841351

So, t = -0.881495 is basically -52.889705962259048106 min,
as in approx -53 min

So, the person has been dead approx 0.881495 hrs ----> ANS

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