3'5 Exponential Growth

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3.5 Exponential Growth DecayModeling Data

• Objectives
• Model exponential growth decay
• Model data with exponential logarithmic
  functions.

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Exponential Growth Decay Models

• A0 is the amount you start with, t is the time,
  and kconstant of growth (or decay)
• f kgt0, the amount is GROWING (getting larger), as
  in the money in a savings account that is having
  interest compounded over time
• If klt0, the amount is SHRINKING (getting
  smaller), as in the amount of radioactive
  substance remaining after the substance decays
  over time

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Graphs
A0
A0
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Could the following graph model exponential
growth or decay?

• 1) Growth model.
• 2) Decay model.

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Example

• Population Growth of the United States. In 1990
  the population in the United States was about 249
  million and the exponential growth rate was 8
  per decade. (Source U.S. Census Bureau)
• Find the exponential growth function.
• What will the population be in 2020?
• After how long will the population be double what
  it was in 1990?

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Solution

• At t 0 (1990), the population was about 249
  million. We substitute 249 for A0 and 0.08 for k
  to obtain the exponential growth function.
• A(t) 249e0.08t
• In 2020, 3 decades later, t 3. To find the
  population in 2020 we substitute 3 for t
• A(3) 249e0.08(3) 249e0.24 ?
  317.
• The population will be approximately 317
  million in 2020.

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Solution continued

• We are looking for the doubling time T.
• 498 249e0.08T
• 2 e0.08T
• ln 2 ln e0.08T
• ln 2 0.08T (ln ex x)
• T
• 8.7? T
• The population of the U.S. will double in about
  8.7 decades or 87 years. This will be
  approximately in 2077.

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Interest Compound Continuously

• The function A(t) A0ekt can be used to
  calculate interest that is compounded
  continuously.
• In this function
• A0 amount of money invested,
• A balance of the account,
• t years,
• k interest rate compounded continuously.

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Example

• Suppose that 2000 is deposited into an IRA at an
  interest rate k, and grows to 5889.36 after 12
  years.
• What is the interest rate?
• Find the exponential growth function.
• What will the balance be after the first 5 years?
• How long did it take the 2000 to double?

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Solution

• At t 0, A(0) A0 2000. Thus the exponential
  growth function is
• A(t) 2000ekt. We know that A(12)
  5889.36. We then substitute and solve for k
• 5889.36 2000e12k
• 
  
• 
  

The interest rate is approximately 9.5
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Solution continued

• The exponential growth function is
• A(t) 2000e0.09t.
• The balance after 5 years is
• A(5) 2000e0.09(5)
• 2000e0.45
• ? 3136.62

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Solution continued

• To find the doubling time T, set A(T) 2 ? A0
  4000 and solve for T.
• 4000 2000e0.09T
• 2 e0.09T
• ln 2 ln e0.09T
• ln 2 0.09T
• T
• 7.7 ? T
• The original investment of 2000 doubled in
  about 7.7 years.

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Growth Rate and Doubling Time

• The growth rate k and the doubling time T are
  related by
• kT ln 2
• or
• or
• The relationship between k and T does not
  depend on A0.

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Example

• A certain towns population is doubling every
  37.4 years. What is the exponential growth rate?
• Solution

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Exponential Decay

• Decay, or decline, is represented by the function
  A(t) A0ekt, k lt 0.
• In this function
• A0 initial amount of the substance,
• A amount of the substance left after time,
• t time,
• k decay rate.
• The half-life is the amount of time it takes for
  half of an amount of substance to decay.

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Example

• Carbon Dating. The radioactive element
  carbon-14 has a half-life of 5715 years. If a
  piece of charcoal that had lost 7.3 of its
  original amount of carbon, was discovered from an
  ancient campsite, how could the age of the
  charcoal be determined?
• Solution The function for carbon dating is
• A(t) A0e-0.00012t.
• If the charcoal has lost 7.3 of its
  carbon-14 from its initial amount A0, then
  92.7A0 is the amount present.

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Example continued

• To find the age of the charcoal, we solve the
  equation for t
• The charcoal was about 632 years old.