ISP 120 Week 2

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ISP 120 Week 2

• Exponential Models

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Activity 1

• Cars and Trucks by Year 2003
• a. Make an X-Y scatter plot of the data. Include
  it in your Word document. 

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• b. Add a trendline to your chart in a. Include
  the equation for the trendline and the R-squared
  value. Paste this chart in your Word document.

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• c. Predict how many cars and trucks there will be
  in the US in the year 2008 using both the
  equation method and the extended trendline
  method. 
• i.  Equation Method  Substitute 2008 for x into
  the equation displayed on your graph in b) and
  solve for y.  Write your answer in your Word
  document.  (In some cases you might need to
  substitute a value for y and solve for x).  Type
  your result in your Word document.
• y 3.4659x - 6718.8
• y 3.4659(2008) - 6718.8
• y 6959.5272 - 6718.8
• y 240.7272 240.73 million
• Therefore, according to the trendline, there will
  be 240.73 million cars and trucks in the US by
  2008.

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• ii.  Extended Trendline Method  Extend the
  trendline on your graph using by right clicking
  on the trendline (you created in b) and then
  choose Options and Forward Forecast.  Enter 5 to
  predict the number of passenger cars and trucks
  there will be in 2008.  This gives you an
  approximate prediction.  Paste the graph in your
  Word document.  Paste the graph in your Word
  document.Always verify the your results match! 
  If not, please see your instructor!

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• d. Use the equation method to predict when there
  were no passenger cars and trucks in the US. Type
  you result in your Word document.  Verify your
  answer by extending the trendline and pasting the
  resulting graph in your Word document.  How
  accurate is the model in this prediction?  In 2-3
  sentences justify whether or not the equation
  should be used to predict when there were no
  passenger cars and trucks.
• y 3.4659x - 6718.8
• 0 3.4659x - 6718.8
• 6718.8 3.4659x
• 1938.544 x
• x 1938

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• Therefore, according to the equation and the
  extended trendline, there were NO cars in 1938.
  However, practical knowledge tells us this cannot
  be accurate because cars were invented in the
  1800s.

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• e. Use both methods to predict the number of cars
  in US in 2050.  Type your answer and paste your
  graph in your Word document. Be sure your results
  match!  Do you think your answer is a good
  estimate? In 2-3 sentences justify whether or not
  the equation should be used to predict how many
  passenger cars and trucks there will be in 2050.
• y 3.4659(2050) - 6718.8
• y 7105.095 - 6718.8
• y 386.295 386.30 million

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• Therefore, according to the equation and the
  trendline, there will be approximately 386.30
  million cars and trucks in 2050. Because the R2
  value is close to 1 and we would expect the
  number of passenger cars and trucks to continue
  increasing, 386.30 million is probably a pretty
  good prediction for 2050.

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• Coal Consumption 2003
• a. Make an X-Y scatterplot of the data. Add a
  trendline, including the equation of the line and
  the R-squared value. Paste this chart into your
  Word document.

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• b. Use both methods (see above) to predict coal
  consumption in the year 2008.  Type you answer
  and paste your graph in you Word document.
• y 0.3374x - 652.75
• y 0.3374(2008) - 652.75
• y 24.7492 24.75 quads

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• c. According to your model, when will the coal
  consumption be 30 quads?
• 30 0.3374x - 652.75
• 30 652.75 0.3374x
• 682.75 0.3374x
• 2023.563 x
• x 2024

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• d. In a short well-written paragraph, discuss
  social, political, or physical changes which
  might affect the accuracy of predictions using
  this model.
• Increase/decrease in usage of other energy
  sources
• Limiting resources/advancing technology
• Development of more efficient means for energy
• Political laws regulating the excavation of coal
• Social reforms effecting labor conditions
• Engagement of war
• Clean air laws

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WEEK 2 Overview

• Introduce Exponential Modeling
• Discuss Radioisotope Dating
• Solving Exponential Equations with Logarithms
• Activities 2 and 3
• Assignment 2

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Exponential Change

• A linear relationship has a fixed rate of change
• An exponential relationship is one in which for a
  fixed change in x, there is a fixed percent
  change in y
• The percent change must be constant for the
  relationship to be exponential
• REMINDER To calculate the actual percent, you
  must multiply your results by 100 to convert to
  percent OR in Excel click the icon on the
  toolbar

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Examples
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General Equation

• y P(1r)x
• P is the starting value (value of y when x 0)
• r is the percent change (written as a decimal)
• x is the input variable (usually time)
• The equation for the previous example when r
  -.5 would be
• y 192(1-.5)x
• y 192.5x

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Where do we encounter exponential relationships?

• Populations tend to grow exponentially
• When an object cools, the temperature decreases
  exponentially toward the ambient temperature
• Radioactive substances decay exponentially
• Bacteria populations grow exponentially
• Money in a savings account with a fixed rate of
  interest
• Viruses and even rumors spread exponentially
  through a population (at first)
• Anything that doubles, triples, halves over a
  certain amount of time
• Anything that increases or decreases by a percent

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Increasing/Decreasing a Number by a Percent

• N P Pr
• N P(1r)
• N is the ending value
• P is the starting value
• r is the percent (in decimal form)
• to convert from percent to decimal form, divide
  by 100 (move the decimal 2 places to the left)
• These two formulas are the same by the
  distributive law

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Examples

• Increase 50 by 10
• Sales tax is 9.75. You buy an item for 37.00.
  What is the final price of the article?
• In 1999, the number of crimes in Chicago was
  231,265. Between 1999 and 2000 the number of
  crimes decreased by 5. How many crimes were
  committed in 2000?

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Exponential Growth Over Time

• Exponential growth is increasing by the same
  percent over and over
• If a quantity P is growing by r each year, after
  one year there will be P(1r)
• After 2 years, we multiply the amount after one
  year by (1r) P(1r)(1r) P(1r)2
• After 3 years, we multiply the amount after 2
  years by (1r) P(1r)2(1r) P(1r)3

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• Therefore, after t years the formula will be
• N P(1r)t
• If the quantity P is decreasing by r, then the
  formula will be
• N P(1-r)t

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Example By Hand

• A bacteria population is at 100 and is growing by
  5 per minute.
• How many bacteria cells are present after 3
  minutes? one hour?
• How many minutes will it take for there to be
  1000 cells?

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Same Example By Excel

• 5 minutes

• 60 minutes

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Another Example w/ Excel

• Country A had population of 125 million in 1995.
  Its population was growing 2.1 a year. Country
  B had a population of 200 million in 1995. Its
  population was decreasing 1.2 a year. What are
  the populations of the countries this year? Has
  the population of country A surpassed the
  population of B? If not, in what year will the
  population of country A surpassed the population
  of B?

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Radioisotope Dating

• Radioactivity is the emission of energy
  (particles) from the nuclei of atoms
• It was discovered in 1896 by Henri Becquerel and
  in 1898 by Marie and Pierre Curie (uranium)

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• The key feature of radioactivity that makes it so
  fascinating is that the energy released is
  enormous compared to typical chemical energies. 
  The typical energy release in the explosion of
  one nucleus of one atom is about a million times
  greater than in a chemical explosion of a single
  atom.

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• Nature of radioactivity
• Their radiation affects the emulsion on a
  photographic film
• Their radiation make certain compounds fluoresce
• Their radiation have special physiological
  effects
• Alexander Litvinenko
• They undergo radioactive decay.
• The atoms of all radioactive elements continually
  decay into simpler atoms and simultaneously emit
  radiations.
• Half-life is the length of time during which, on
  the average, half of a given number of atoms of a
  radioactive nuclide decays.
• For example, the half-life of radium-226 is 1620
  years. This means that one-half of a given
  sample of radium-226 can be expected to decay
  into simpler atoms in 1620 years.

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• The splitting of heavy nucleus into nuclei of
  intermediate mass is called fission, during which
  neutrons are emitted and a large amount of energy
  is released
• In a fission reaction, the target nucleus is
  bombarded with neutrons and may break up in a
  great many different ways
• In most cases, two or three neutrons are emitted
  in the process

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Chain Reaction
A chain reaction is a reaction in which the
material or energy that starts the reaction is
also one of the products and can cause similar
reactions.
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• The first controlled chain reaction was actually
  carried out here in Chicago at the University of
  Chicago in 1942 by physicist Enrico Fermi (Nobel
  Prize 1938)
• Nuclear Energy sculpture by Henry Moore on
  Ellis Avenue between 55th and 57th Streets

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Nuclear Power

• A nuclear reactor is a device in which the
  controlled fission of certain substances is used
  to produce new substances and energy
• A nuclear reactor is used in power production at
  nuclear power plants
• The nuclear reactor must be able to sustain a
  chain reaction

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Radioisotopes

• Isotopes of an element have nuclei with the same
  number of protons but different numbers of
  neutrons
• A radioisotope is a radioactive isotope of an
  element
• Some radioisotopes are the products of natural
  radioactive decay
• A far greater number of radioisotopes are
  prepared artificially by the nuclear reactor
  during the fission process

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Radioisotope Dating

• A process that employs the use of a natural
  radioisotope to learn the age of an ancient
  object
• Example
• The Dead Sea Scrolls have about 78 of the
  normally occurring amount of Carbon 14 in them. 
  Carbon 14 decays at a rate of about 1.202 per
  100 years.

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• 2007 2000 7 AD
• 2007 2100 93 BC
• Current estimates are at a 95 confidence
  interval for their date to range from 150 BC to 5
  BC

http//www.lbl.gov/abc/wallchart/chapters/01/overv
iew.html
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Solving Exponential Equations
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Solving for Time

• To solve an exponential equation algebraically
  for time, we must use logarithms
• Remember that what you do to one side of an
  equation, you must do to the other side as well

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Problem 1

• A bacteria population is at 100 and is growing by
  5 per minute. How long will it take for the
  number of cells in the dish to reach 1000?
• Use logarithms to find an EXACT answer.

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Problem 2

• If you deposited 100 into a savings account that
  grows at 3.4 compounded annually, how long will
  it take for your balance to double?

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Problem 3

• Lets revisit the bacteria situation above.
  Assuming we allowed the bacteria population to
  reach 3000 then put in an antibiotic that killed
  the cells at a rate of 22.5 a minute, how long
  would it take for the population to decline to 60
  cells?

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