Anatomy of a Rumour

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Anatomy of a Rumour
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Question

• Could we determine a function S such that S(t)
  approximates the number of students that know the
  rumour at a time arbitrary time t, where t is
  measured in, say, hours?

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Activity

• Students are given small pieces of paper with the
  numbers 1 to 10 written in a column. Next to the
  number 1, one student had an R, meaning that the
  student knew the rumour. Each student was told to
  show their paper to one other student during each
  round. If the student saw an R on any of the
  papers that they saw, they then knew the rumour
  and wrote an R next to the number of that round.
  The rounds continued until all students knew the
  rumour.

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• Do the activity twice
• Once with a very shy student having the rumour
  and once with the out there student having the
  rumour.

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Decide what the graph of S might look like
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Data
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Spread of rumour
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Class Results
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Describe three conditions that dS/dt, the rate of
spread of the rumour, should satisfy.

• Keep in mind that we are describing the rate of
  change of the number of students who know the
  rumour.
• Consider the nature of the rumour itself,
  conditions at your school.
• Decide at least one condition that changes as the
  rumour spreads

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Some possible factors

• How "juicy" the rumour is. The juicier the rumour
  the faster it will spread.
• How many people know the rumour at the given
  time. The more that know the faster it spreads.
• The average number of contacts a student makes
  with other students. The more contacts the faster
  the rumour spreads.

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Some possible factors

• The number of students who do not know the
  rumour. Early on, almost everyone will be hearing
  the rumour for the first time. Later, after the
  rumour has had time to spread, many students will
  be reporting the rumour to those "already in the
  know." Even though the average number of contacts
  a student makes with other students remains
  constant, the rate of spread of the rumour begins
  to decrease because fewer students are hearing
  the rumour for the first time.

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The rate of spread of the rumour is proportional
to

• The number of students who know the rumour, S
• the number of students who do not know the
  rumour, M - S
• M is the total number of students in the school

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The symbolic description dS/dt that incorporates
the conditions that we listed above
k is a constant that depends on both the
juiciness of the rumour the average number of
contacts between students.
Remember that the symbols S and dS/dt are not
constants, but functions of t.
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The symbolic description dS/dt that incorporates
the conditions that we listed above
k is a constant that depends on both the
juiciness of the rumour, the average number of
contacts between students.
We assume that k increases with juiciness and
with an increase in the average number of
contacts.
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According to our differential equation
under what circumstances will the rate
of spread of the rumour will be 0?
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If the units for dS/dt are students per hour,
what units should be assign to the constant k?
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Some questions to pose

• Compare the rate at which the rumour spreads when
  1/4 of the students know the rumour, i.e.,
• S (1/4)M, to the rate when 3/4 know it.
• Is the rumour spreading faster when 1/2 of the
  students know the rumour or when 3/4 know the
  rumour?
• What happens to dS/dt as S(t) approaches M? Why
  is this reasonable?

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Initial condition

• In the beginning 2 people knew the rumour. This
  means that S(0) 2
• We can find a symbolic description for the
  solution of the problem.

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How could we verify that this is the solution to
our differential equation?
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Salt Questions
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Question 1

• A tank contains 1000 litres of brine with 15kg of
  dissolved salt. Pure water enters the tank at a
  rate of 10 litres/min. The solution is kept
  thoroughly mixed and drains from the tank at the
  same rate. How much salt is in the tank after t
  minutes?

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Define the function

• s(t) kg of salt in the tank at time t minutes
• This is what we are looking for.
• S(t) the rate at which the amount of salt in
  the tank is changing
• (rate of salt going in) - (rate of salt
  going out)

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Rate of salt going into the tank

• This is 0
• This is expected because pure water is flowing in

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Rate of salt leaving the tank
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Solving
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• When t 0, 15kg of dissolved salt was in the
  tank.

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Question 2

• The graph below shows y (a - x)(x - 2a) and y
  a - x where a is a positive constant.
• Find the value of a so that the area between the
  two functions is divided above and below the
  x-axis in a ratio of 11.

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Graph
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Allow students time to get deeply into a lot of
working and then simplify the problem for them
using transposition.
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Question 2d
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Question 2d
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Question 2d
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Question 2d
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Question 3

• Use as many different methods that you can think
  of to find the area between the parabola
• and the line

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Extending knowledge of integrals.

• Volumes of revolution around x and y axis
• Shell rule
• Equation for length of a curve
• Surface area

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Question 4