Lesson 9-NC

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Lesson 9-NC

• Newtons Law of Cooling

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Objectives

• Use Newtons Law of Cooling to solve problems

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Vocabulary

• Initial condition allows the user to find the
  particular solution from a family of solutions
• Equilibrium a steady state condition with
  neither growth nor decay

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Temperature Change

• An objects temperature over time will approach
  the temperature of its surroundings (the medium)
• The greater the difference between the objects
  temperature and the mediums temperature, the
  greater the rate of change of the objects
  temperature
• This change is a form of exponential decay

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CSI Newtons Law of Cooling

• The rate at which an object cools is proportional
  to the difference in temperature between the
  object and the surrounding medium
• Where T is temperature of the object,
• k is a proportionality constant,
• M is the temperature of the
  surrounding medium
• and t is time
• A coroner uses this to help determine the time of
  death and is seen in every Crime TV series from
  Dragnet to CSI.

dT ---- k(T M) dt
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Newtons Cooling Equation

• Given
• by changing the variable T to y(t) T M we get
  the following equation
• a very familiar differential equation, whose
  solution is
• changing back to T, we get T(t) Tm (T0
  Tm)ekt
• where k will always be negative (from decay)

dT ---- k(T M) dt
dy ---- k(y) dt
y(t) y0ekt
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Example
Example A potato is taken out of a 300o F
oven and left to cool in a room at 75o F. Write
a differential equation expressing the change in
rate of the temperature of the potato, T, with
respect to time, t.
dT ---- k(T M) dt
dT ---- k(300 75) dt
dT ---- 225k dt
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Example cont
Example A potato is taken out of a 300o F
oven and left to cool in a room at 75o F. Write
a differential equation expressing the change in
rate of the temperature of the potato, T, with
respect to time, t.
dTo ---- k(To Tm) dt T(t) Tm (To
Tm)e kt T(t) 75 (300 75)e kt T(t)
75 225e kt
Use intermediate condition to find k
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Newtons Law of Cooling
Example The great detective Sherlock Holmes
and his assistant, Dr. Watson, are discussing the
murder of actor Cornelius McHam. McHam was shot
in the head, and his understudy, Barry Moore, was
found standing over the body with the murder
weapon in hand. Lets listen in Watson Open-and
-shut case, Holmes. Moore is the
murderer. Holmes Not so fast, Watson you are
forgetting Newtons Law of Cooling! Watson Huh?
Holmes Elementary, my dear Watson. Moore was
found standing over McHam at 1006 p.m., at which
time the coroner recorded a body temperature of
77.9F and noted that the room thermostat was set
to 72F. At 1106 p.m. the coroner took another
reading and recorded a body temperature of
75.6F. Since McHams normal temperature was
98.6F, and since Moore was on stage between 600
p.m. and 800 p.m., Moore is obviously innocent.
Ask any calculus student to figure it out for
you. How did Holmes know that Moore was innocent?
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CSI Solution
T(t) Tm (To Tm)e kt T(t) is temperature
of the body at t hours since death Tm 72
temperature of the room T0 98.6 temperature of
the body t would represent the hours since
death But we dont know the time of death. We
can use the coroner's temperature readings to
determine k. T(1006) 77.9 T(1106) 75.6
so T(1) 75.6 72 (77.9 72)ek
3.6
5.9ek
k -ln(3.6/5.9) .494019 T(t0)
98.6 T(t, 1006) 77.9 72 26.6e-0.494019t 5.9
26.6e-0.494019t t ln(5.9/26.6) / -0.494019
3.048 hours since death so death occurred at
about 7 pm.
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Summary Homework

• Summary
• Newtons Law of Cooling has a wide variety of
  uses
• Homework
• pg 621 14, 15, 16