Exponential Growth and Decay, Part II

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Exponential Growth and Decay, Part II

• Dr. Dillon
• Calculus II
• SPSU
• Fall 1999

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Todays Goals

• Identify common examples of growth and decay
  problems
• Solve growth and decay problems

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Another Population Problem

• Switzerlands population grows exponentially at a
  rate of 0.2 per year.
• Its 1988 population was 6.6 million.
• Find an expression for the population as a
  function of time in years.

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The Solution?
In our basic model
the rate of growth is k
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Whered You Get That?

• Translate the phrase, grows... at a rate of 0.2
  a year and you get

The growth rate is 0.2 of what?
0.2 of the current population, P(t).
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The Answer
Where P(t) is measured in millions
t is measured in years since 1988
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Another...

• Money in a bank account grows continuously at an
  annual rate of r.
• In 1992, 2000 is put in that account.
• Find the interest rate if the account makes
  341.16 in interest by the end of 1999.

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The Model

• where P(t) is the number of dollars in the
  account t years after 1992 and r is the interest
  rate

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What Do We Know?

• When t7,
• P(t)2341.16

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What Do We Want?

• Solve for r when

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Thats Easy!

• Take the natural logarithm of both sides of the
  equation to solve for r.

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Finally

• Using the calculator, we find r0.0225

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So the Interest Rate is...

• 2.25

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Newtons Law of Cooling

• The rate of cooling of an object is directly
  proportional to the temperature difference
  between the object and its surroundings.

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The Problem

• If an object takes 40 minutes to cool from 30
  degrees to 24 degrees in a 20 degree room, how
  long will it take the object to cool to 21
  degrees?

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Solving the Diff Eq
Same as before?
Not exactly, but close.
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Exponentiate Both Sides ...
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C?
So C is the difference between the initial
temperature of the cooling body and the
surrounding room.
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So in This Case?

• Since the initial temperature of the body is 30
  degrees, we have

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k?

• Since it takes 40 minutes for the body to cool
  from 30 degrees to 24 degrees we have

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Thus
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How Long to Cool to 21?

• We want t when f(t)21 so

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To Finish...