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Newton’s Law of Cooling
UBC Math 102
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)
Office hrs: MX 1111
• Office hrs this week (None today) • Thurs 12:00-‐1:30pm • Friday 11:30-‐12:30
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
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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
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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
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
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
Total mass (to be op`mized):
Largest mass: find `me at which M’(t)=0
UBC Math 102
Best prac`ce
• Consider first working with `dy formula
• Before plugging in your numbers
UBC Math 102
Cri`cal point (best harvest `me)
0 = M’(t) Where is the half-‐life of the cricket popula`on.
Quiz 3 groupà individual [Q1]
The same func`on! Cri`cal pts:
UBC Math 102
Quiz 3 – [Q2]
Rectangle with fixed diagonal length L=1 Maximize area of rectangle
[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
[Q3] Spherical balloon
If the volume of a spherical balloon increases at a constant rate then its radius increases at
UBC Math 102
Solu`on to [Q3]:
Given: Hence: à à
UBC Math 102
It was a dark and stormy night
• hcps://www.youtube.com/watch?v=RinnIWFgF7w
UBC Math 102
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.
It was a dark and stormy night. In one dark alley, a solitary figure walked alone, hardly realizing that danger lurked around the corner…
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
The body was discovered at 2AM
2:00 am
The Temperature of the body was 21o C
21
One hour later the body had cooled down to 17o C
3:00 am
17
Just before the murder, the body temperature was 37o C.
But at the instant of death, he started cooling off.
T
t
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
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
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
(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
Related to the class of differen`al equa`on of form:
Which we know, has a solu`on:
UBC Math 102
Recall behaviour of this func`on:
UBC Math 102
t
y(t)
y0
1
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
(2) Newton’s Law of Cooling `me behaviour:
The graph of the func`on looks like:
UBC Math 102
Solu`on
UBC Math 102
The temperature will approach a steady state
UBC Math 102
Steady state
(3) Back to solving a murder
To solve the murder mystery, we will use the predicted temperature
UBC Math 102
The `me line
Our goal is to find the `me of death (ToD)
The data:
Time Temperature Comments:
UBC Math 102
Calcula`on:
Use the temperature fn: Rewrite: Use the data: Simplify:
UBC Math 102
Calcula`on, cont’d:
1 hr later:
UBC Math 102
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
(4) Finding k
The equa`ons are Then the value of k is:
UBC Math 102
Value of k:
We find:
UBC Math 102
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
(5) Time of death
The `me of death is
UBC Math 102
Solu`on:
UBC Math 102
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
(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
Prac`ce problems
UBC Math 102
(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
(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.
Prac`ce test problem:
Prac`ce test problems
UBC Math 102
Prac`ce exam ques`on:
UBC Math 102
Solu`on:
UBC Math 102
Solu`ons to prac`ce exam problems from last `me
UBC Math 102
Prac`ce exam ques`on (1):
UBC Math 102
Solu`on:
UBC Math 102
Solu`on, cont’d:
Prac`ce exam ques`on (2)
UBC Math 102
Solu`on part (a):
UBC Math 102
Solu`on part (b):
UBC Math 102
Prac`ce exam ques`on (3):
UBC Math 102
Solu`on part (a):
UBC Math 102
Solu`on part (b)
UBC Math 102
Answers
quiz 3 [Q3] -‐ E 1 E 2 C 3 C 4 B 5 B 6 B
UBC Math 102
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