Modeling with Exponentials

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Modeling with Exponentials Logarithms

• Lecture Notes
• Precalculus Math 1113
• September 8, 2005

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Example 1

• Solve the following equation

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Example 2 Pg. 374 68

• A learning curve is a graph of P(t) that shows
  performance as a function of time spent training.
  We model learning curves with P(t) M C e-kt,
  with k and C positive and C lt M.
• Draw a graph of P(t).
• For a pole vaulter, the learning curve is given
  by P(t) 20 14 e-.024t, where t is months of
  training and P(t) is height in feet one can
  vault. After how many months is he able to vault
  12 ft?

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Example 3

• Exponential Growth
• E. Coli bacteria live in animal intestines, grow
  rapidly and can be harmful to humans, especially
  children. The number of E. Coli can double every
  49.5 minutes. Suppose there are initially
  500,000 E. Coli bacteria per millimeter of
  intestine.
• How many bacteria will there be per millimeter
  after 3 hours?
• At what time will the number of bacteria exceed
  25 million?

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Example 4 Pg. 388 22

• Radium-226 Dating Methods
• The half-life of radium-226 is 1600 years.
  Suppose we have a 22-mg sample.
• Find a function that models the mass remaining
  after t years.
• How much of the sample will remain after 4000
  years?
• After how long will only 18 mg remain?
• How long will it take for 95 of the sample to
  decay?

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Example 5 Pg. 387 10

• Bacteria Exponential Growth
• The count in a culture of bacteria was 400 after
  2 hours and 25,600 after 6 hours.
• What is the relative rate of growth expressed as
  a percentage?
• What was the initial size of the culture?
• Find a function that models the number of
  bacteria n(t) after t hours.
• Find the number of bacteria after 4.5 hours.
• When will the the number of bacteria be 50,000?

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Example 6 Pg. 388 26

• Newtons Law of Cooling
• A kettle full of water is brought to a boil in a
  room with ambient temperature 20o C. After 15
  minutes, the temperature of the water has
  decreased from 100o C to 75o C.
• Find the temperature after another 10 minutes.
• Graph the cooling function
• When will the temperature be less than 22o C?

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Logarithmic Scales

• We use logarithmic scales for
• Sound (Decibels)
• Earthquakes (Richter)
• Acids vs. Bases (pH scale)
• Why?
• Keeps huge range of numbers similar
• Perfect for phenomena with threshold values below
  which no effect is detected

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Logarithmic Scales

• The following formulas will be given on any test
  or exam
• Sound
• Earthquakes
• Acidity

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Example 7 Pg. 388 30

• Acidity
• The pH reading of a a glass of liquid is given.
  Calculate the hydrogen ion concentration of the
  liquid.
• Beer pH 4.6
• Water pH 7.3

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Example 8 Pg. 389 32

• Acidity
• The pH readings of wines vary from 2.8 to 3.8.
  Give the range of hydrogen ion concentrations.

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Example 9 Pg. 389 40

• Sound Levels
• The noise from a lawn mower was measured at 106
  dB. The Nirvana concert was measured at 120 dB.
  Jackhammer noise measured at 132 dB.
• How many times louder was the concert than the
  lawn mower?
• How many times louder was the jackhammer than the
  concert?
• How many times more annoying was Nirvana than the
  jackhammer?

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Example 10

• Earthquakes
• The evening news reported about a California
  earthquake that was 35 times more intense than
  one that measured 6.5 on the Richter scale. What
  was the Richter scale value of the quake?
• Two famous earthquakes hit San Francisco in the
  20th century, the 1906 quake (R 8.25) and the
  1989 World Series quake (R 7.1). The most
  severe earthquake ever recorded hit Japan in 1933
  (R 8.9)
• How many times more intense was the 1906 quake
  than the World Series quake?
• How many times more intense was the Japan quake
  than the 1906 quake?