Exponential Systems

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Exponential Systems
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Exponential systems
Radioactive Decay, population growth, interest
earned on a bank account

• We have talked about Exponential Systems before.
  
• Imagine a system where change is dependant on the
  amount present. That is,
• Where t is time and k is a constant.
• If we know the initial condition y y0, then we
  can find an equation for y based on any t value.

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If we know the initial condition (when t 0)
then we can determine A
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Exponential systems

• All systems that grow/shrink exponentially fit a
  general pattern
• Where y0 is the initial amount, k is a constant
  that will often have to be determined and t is
  time.

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• The rate at which the number of cookies present
  at C lunch changes is proportional to the number
  of cookies present.
• Time is measured in minutes. If there are 140
  cookies on the first tray out (t 0) how many
  will be present after 5 minutes.
• THAT IS, when t 5

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A new problem

• The tray of 140 cookies is put out at E lunch.
  After 2 minutes, only 17 cookies remain. How
  many will be left after 5 minutes?

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What was seen here?

• In Initial/Later condition problems you will need
  to find k to continue the problem. This is
  typical in AP problems.

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Another Problem

• A small amount of radioactive substance is placed
  in a lead container. After 24 hours it is
  observed that 6/7 of the substance remains. If
  the rate of decay is proportional to the amount
  of substance present at any time, what is the
  half life? How long will it take for it to be
  reduced to 1/5 of the original amount?

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Homework

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