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Newton’s Law of Cooling UBC Math 102

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Page 1: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Newton’s  Law  of  Cooling  

UBC    Math  102  

Page 2: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Coming  up  (this  week)  

   

UBC    Math  102  

No  classes  

Hand  in  OSH  by  12:30  in  MX  

1111  

Come  to  class  early  for  

some  exam-­‐type  review    (starts  at  8:30)  

Page 3: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Office  hrs:  MX  1111  

•  Office  hrs  this  week              (None  today)    •  Thurs  12:00-­‐1:30pm  •  Friday  11:30-­‐12:30      

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Results  of  Anonymous  survey  

UBC    Math  102  

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Which  is  useful?    (a)  class  lectures  (b)  Videos  and/or  prelect  Qs  (c)  Wiki  site  (d)  Math  102  Course  Notes  (e)  Webwork  and/or  OSH    (f)  Piazza  (g)  Instructor’s  office  hours  (h)  Posted  slides      (i)  High  school  notes/  book  (j)  Friends,  other  students  (k)  Other    

Page 5: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

How  many  students  use  how  many  resources:  

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(a)  class  lectures  (b)  Videos  and/or  prelect  Qs  (c)  Wiki  site  (d)  Math  102  Course  Notes  (e)  Webwork  and/or  OSH    (f)  Piazza  (g)  Instructor’s  office  hours  (h)  Posted  slides      (i)  High  school  notes/  book  (j)  Friends,  other  students  (k)  Other    

Page 6: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Sugges`ons  for  improvement  

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Did  you  know?  •  There  are  lots  of  prac`ce  final  exams:  •  hcp://www.math.ubc.ca/Ugrad/pastExams/index.shtml  •  And  exam  resource  site  with  hints  and  solu`ons:  •  hcp://wiki.ubc.ca/Science:Math_Exam_Resources  •  hcp://merapp.herokuapp.com/  •  Interac`ve  search  by  topic  or  by  exam/year  

UBC    Math  102  

Page 8: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

More  prac`ce  problems:  (1) Try  out  the  problems  on  doc.  camera  before  the  

lecture  every  class  –  this  is  your  chance  to  prac`ce  and  get  the  idea  before  I  go  over  it  in  class.  

(2) Check  out  the  extra  examples  at  the  end  of  each  lecture  

(3) Check  out  the  Course  Notes  Exercises  and  Review  problems  –    I  have  solu`ons  for  those  of  you  who  try  out  the  problems)  

(4)   Try  the  old  exams  with  the  hints  and  solu`ons  

UBC    Math  102  

Page 9: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Quiz  3  

Cricket  op`mal  harves`ng:    Mass  of  1  cricket  depends  of  `me    (“Consider  a  power  func`on  fit”)    Number  of  crickets  (“decays  exponen`ally  “):            

UBC    Math  102  

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Total  mass  (to  be  op`mized):  

       

Largest  mass:  find  `me  at  which  M’(t)=0  

UBC    Math  102  

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Best  prac`ce  

•  Consider  first  working  with  `dy  formula  

•  Before  plugging  in  your  numbers    

UBC    Math  102  

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Cri`cal  point  (best  harvest  `me)  

     0  =    M’(t)              Where          is  the  half-­‐life  of  the  cricket  popula`on.      

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Quiz  3  groupà  individual  [Q1]  

 The  same  func`on!          Cri`cal  pts:  

UBC    Math  102  

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Quiz  3  –  [Q2]  

Rectangle  with  fixed  diagonal  length  L=1      Maximize  area  of  rectangle  

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[Q2]  cont’d  

Cri`cal  points:            Get  x  =  y  (i.e.  a  square)..  Argue  that  this  is  local  max,  e.g.  using  rough  sketch  of  A  or  other  test.    

UBC    Math  102  

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[Q3]  Spherical  balloon  

If  the  volume  of  a  spherical  balloon  increases  at  a  constant  rate  then  its  radius  increases  at    

UBC    Math  102  

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Solu`on  to  [Q3]:  

   Given:    Hence:                                                                              à                                      à    

UBC    Math  102  

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It  was  a  dark  and  stormy  night  

•  hcps://www.youtube.com/watch?v=RinnIWFgF7w  

UBC    Math  102  

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Humble  pie  

•  …    apologies  that  in  class  the  very  last  step  had  a  careless  error  by  yours  truly,  Detec`ve  Sherlockeshet  Holmes.  (..or  is  it  the  bumbling  Inspector  Cleahseau):  

•  hcps://www.youtube.com/watch?v=64yianfGvzc  

I  have  corrected  the  `me-­‐line  to  make  things  work  out  correctly..  All  other  calcula`ons  were  OK.  

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It was a dark and stormy night. In one dark alley, a solitary figure walked alone, hardly realizing that danger lurked around the corner…

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Murder  Mystery  102  

At  2AM,  the  police  discovered  that  Sir  Edmund  had  been  murdered.  But  when  had  the  murder  been  commiced?  And  which  of  two  suspects  was  responsible??  

UBC    Math  102  

Page 23: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

The body was discovered at 2AM

2:00 am

Page 24: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

The Temperature of the body was 21o C

21

Page 25: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

One hour later the body had cooled down to 17o C

3:00 am

17

Page 26: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Just before the murder, the body temperature was 37o C.

But at the instant of death, he started cooling off.

T

t

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Police  report  •  FACT:  The  temperature  of  the  body  at  the  ToD  (`me  of  death)  was  37oC.  

•  FACT:  When  the  body  was  discovered  (2AM),  its  temperature  was  21  oC.  

•  FACT:  One  hour  later  (3AM),  the  body’s  temperature  was  17oC.  

•  FACT:  The  ambient  temperature  on  Main  St  was  7oC  all  night.  

•  FACT:  Suspect  A  has  a  solid  alibi  un`l  12:00AM  •  FACT:  Suspect  B  has  a  solid  alibi  un`l  11:30PM  

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Take  a  guess:  

Which  suspect  could  have  commiced  the  murder?    (A)  Suspect  A  (B)  Suspect  B  (C)  Either  Suspect  A  or  B  (D)  Neither  one.  (E)  There  is  not  enough  informa`on  to  know.    

UBC    Math  102  

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Newton’s  Law  of  Cooling  

The  rate  of  change  of  the  temperature  T  of  a  body  is  propor`onal  to  the  difference  between  its  temperature  and  the  ambient  temperature,  E.      Let  T(t)  =  temperature  of  the  both  at  `me  t  E  =  Ambient  (environment)  temperature.  

UBC    Math  102  

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(1)  Newton’s  Law  of  Cooling  (NLC)  The  rate  of  change  of  the  temperature  T  of  a  body  is  propor`onal  to  the  difference  between  its  temperature  and  the  ambient  temperature,  E.    

Which  differen`al  eqn  does  T  sa`sfy?  (Assume  k>0)      

UBC    Math  102  

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Related  to  the  class  of  differen`al  equa`on  of  form:  

     Which  we  know,  has  a  solu`on:    

UBC    Math  102  

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Recall  behaviour  of  this  func`on:  

           

UBC    Math  102  

t  

y(t)  

y0  

1  

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Newton’s  Law  of  Cooling    `me  behaviour:  

The  temperature  of  an  object  T(t)  exposed  to  the  ambient  temperature  E  is      What  does  the  graph  of  this  func`on  look  like?      

UBC    Math  102  

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(2)  Newton’s  Law  of  Cooling    `me  behaviour:  

The  graph  of  the  func`on    looks  like:    

UBC    Math  102  

Page 35: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on  

UBC    Math  102  

Page 36: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

The  temperature  will  approach  a  steady  state  

UBC    Math  102  

Steady    state  

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(3)  Back  to  solving  a  murder    

To  solve  the  murder  mystery,  we  will  use  the  predicted  temperature      

UBC    Math  102  

Page 38: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

The  `me  line  

 Our  goal  is  to  find  the  `me  of  death  (ToD)        

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The  data:  

Time          Temperature                          Comments:  

UBC    Math  102  

Page 40: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Calcula`on:  

Use  the  temperature  fn:    Rewrite:    Use  the  data:    Simplify:  

UBC    Math  102  

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Calcula`on,  cont’d:  

1  hr  later:  

UBC    Math  102  

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Using  the  data  to  find  k  

Ater  using  the  given  data      We  arrive  at  the  equa`ons      What  is  the  value  of  k?        

UBC    Math  102  

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(4)  Finding  k  

The  equa`ons  are      Then  the  value  of  k  is:          

UBC    Math  102  

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Value  of  k:  

We  find:  

UBC    Math  102  

Page 45: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Time  of  death  

Now  that  we  have  found  the  constant  k,  we  can  find  the  `me  of  death  t  from  either  equa`on,  for  example,  from      What  `me  of  death  do  we  find?  (Express  in  terms  of  k,  which  is  already  known)    

UBC    Math  102  

Page 46: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

(5)  Time  of  death  

The  `me  of  death  is    

UBC    Math  102  

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Solu`on:  

UBC    Math  102  

     

Page 48: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on:  

UBC    Math  102  

•  The  murder  occurred  2  hrs  and  16  min  before  the  body  was  discovered.  

•  Thus  the  `me  of  death  was  11:44  PM  

Page 49: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

(6)  Who-­‐dunnit?  

Suspect  A  has  an  alibi  `ll  12:00  midnight,  and  Suspect  B  `ll  11:30PM  that  night.  Which  suspect  could  have  commiced  the  murder?      

UBC    Math  102  

Page 50: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  problems  

UBC    Math  102  

Page 51: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

(1)  From  doc.  camera  before  class:  

         Hints:  Find  a,b,  by  matching  the  deriva`ve  of  the  func`on  T(t)  to  the  diff’l  eqn,  and  using  the  ini`al  condi`on.  Soln:  before  class  on  Friday  

Page 52: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

(2)  More  pre-­‐class  prac`ce:  

           Hint:  the  half-­‐life  is  related  how  to  k?  The  ini`al  level  is  y(0)=10A    where  A  =  acceptable  level.  

Page 53: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  test  problem:  

Page 54: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  test  problems  

UBC    Math  102  

Page 55: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  exam  ques`on:  

UBC    Math  102  

Page 56: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on:  

UBC    Math  102  

Page 57: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`ons  to  prac`ce  exam  problems  from  last  `me  

UBC    Math  102  

Page 58: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  exam  ques`on  (1):  

UBC    Math  102  

Page 59: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on:  

UBC    Math  102  

Page 60: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on,  cont’d:  

Page 61: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  exam  ques`on  (2)  

UBC    Math  102  

Page 62: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on  part  (a):  

UBC    Math  102  

Page 63: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on  part  (b):  

UBC    Math  102  

Page 64: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Prac`ce  exam  ques`on  (3):  

UBC    Math  102  

Page 65: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on  part  (a):  

UBC    Math  102  

Page 66: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Solu`on  part  (b)  

UBC    Math  102  

Page 67: Newton’s)Law)of)Cooling) - University of British ...keshet/M102/2015/Lect10.1.pdf · Newton’s)Law)of)Cooling) The)rate)of)change)of)the)temperature)T)of)abody)is) propor‘onal)to)the)difference)between)its)temperature)

Answers  

quiz  3  [Q3]  -­‐  E  1  E  2  C  3  C  4  B  5  B  6  B  

UBC    Math  102  


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