A body was found in a room where the room\'s temperature is 70 degrees F. The ra

Question

A body was found in a room where the room's temperature is 70 degrees F. The rate of temperature change of the body is proportional to the difference between the room temperature and the body temperature. Let y(t) represent the temperature of the body, t hours from the time of death.

(a) Write the differential equation satisfied by y(t).

(b) After several measurements, it was determined that when the temperature of the body was 80 degrees   F, it was decreasing at the rate of 5 Degrees F per hour. Find the constant of proportionality.

(c) Suppose that at the time of death, the body's temperature was about normal, say 98 degrees F. Determine y(t).

(d) When the body was discovered, its temperature was 85 degrees F. Determine how long ago the person died.

Explanation / Answer

(a) Since the rate of temperature change (dy/dt) is proportional to the difference between room temperature and body temperature (y(t)-70), we can relate them with the following equation:

dy/dt = k(y-70), where k is the constant of proportionality

(b) This says that when y(t) = 80, dy/dt = -5  (it is negative because the temperature was decreasing.  Let's plug in and solve for k.

-5 = k(80-70)

-5 = k(10)

k = -1/2

(c) Let's say the person died at t=0.  Then y(0) = 98.  To answer this question, we will have to solve the differential equation and use y(0)=98 as the initial condition.

dy/dt = (-1/2)(y-70) = -y/2 + 35

y' + y/2 = 35

This equation will have a particular solution and a homogeneous solution.  Let's find the homogeneous solution first:

y' + y/2 = 0

if y = Aemt, then y' = Amemt

Amemt + (A/2)emt = 0

Divide both sides by Aemt

m + 1/2 = 0

m = -1/2

So our homogeneous solution is:

yh = Ae-t/2

Our particular solution is just y = 70:

(70)' + 70/2 = 35

So our total solution is the sum of both

y = Ae-t/2 + 70

Now we can use our initial condition

y(0) = 98 = A + 70

A = 28

Our solution is therefore

y = 28e-t/2 + 70

Check: y' = -(28/2)e-t/2

Does it equal -(1/2)[28e-t/2 + 70 - 70] = -1/2[28e-t/2].  They are the same.

(d) How long ago did the person die if their current temperature is 85 degrees?  We just plug and chug:

y(t) = 85 = 28e-t/2 + 70

15 = 28e-t/2

e-t/2 = 15/28

-t/2 = ln(15/28)

t = -2*ln(15/28)

t = 1.25 hours = 1 hour, 14 minutes, 54 seconds

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