copyright © 2011 pearson education, inc. modeling our world 9b discussion paragraph 1 web 50....

Copyright © 2011 Pearson Education, Inc.

Modeling Our World

9B Discussion Paragraph

1 web50. Alcohol Metabolism51. Property Depreciation

1 world52. Linear Models53. Nonlinear Models

Copyright © 2011 Pearson Education, Inc. Slide 9-3

Unit 9C

Exponential Modeling

9-C


Exponential Functions

An exponential function grows (or decays) by the same relative amount per unit time. For any quantity Q growing exponentially with a fractional growth rate r,

Q = Q0 (1+r)t

whereQ = value of the exponentially growing quantity at time tQ0 = initial value of the quantity (at t = 0)

r = fractional growth rate for the quantityt = time

Negative values of r correspond to exponential decay.Note that the units of time used for t and r must be the same.

US Population GrowthCN (1a-b)

The 2000 census found a US population of about 281 million, with an estimated growth rate of .7% per year.

a. Write an equation for the US population that assumes exponential growth at this rate.

b. Use the equation to predict the US population in 2100.


Declining PopulationCN (2a-b) China’s one child policy was originally

implemented with the goal of reducing China’s population to 700 million by 2050. China’s 2009 population was about 1.3 billion. Suppose china’s population declines at a rate of .5% per year.

a. Write and equation for the exponential decay of

the population. b. Will this rate of decline be sufficient to meet the

original goal?


9-C


To graph an exponential function, use points corresponding to several doubling times (or half-lives, in the case of decay). Start at the point (0,Q0), the initial value at t = 0.

For an exponentially growing quantity, the value of Q is 2Q0 (double the initial value) after one doubling time (Tdouble), 4Q0 after two doubling times (2Tdouble), 8Q0 after three doubling times (3Tdouble), and so on.

For an exponentially decaying quantity, the value of Q falls to Q0/2 (half the initial value) after one half-life (Thalf), Q0/4 after two half-lives (2Thalf), Q0/8 after three half-lives (3Thalf), and so on.

Graphing Exponential Functions

9-C


Exponential Growth

To graph exponential growth, first plot the points (0,Q0), (Tdouble,2Q0), (2Tdouble,4Q0), (3Tdouble,8Q0), and so on. Then fit a curve between these points, as shown tothe right.

9-C


Exponential Decay

To graph exponential decay, first plot the points (0,Q0), (Thalf,Q0/2), (2Thalf,Q0/4), (3Thalf,Q0/8), and so on. Then fit a curve between these points, as shown to the right.

9-CSensitivity to Growth RateCN (3a-b) The growth rate of the US population has varied

substantially during the past century. It depends on the immigration rate, as well as birth and death rates.

a. Starting from the 2000 census, which found a population of 281 million, project the population using the 2100 using growth rates that are .2 percentage point lower and higher than the .7% used in problem 1.

b. Make a graph showing the population through 2100 for each growth rate.


9-C


If given the growth or decay rate r, use the form

If given the doubling time Tdouble, use the form

If given the half-life Thalf, use the form

Forms of the Exponential Function

9-CMontly and Annual Inflation RatesCN (4a-b)

The US government reports the rate of inflation (measured by CPI) both monthly and annually.

Suppose that, for a particular month, the monthly rate of inflation is reported as .8%.

a. What annual rate of inflation does this imply? b. Is the annual rate 12 times the monthly rate?

Explain.


9-C


China’s Coal ConsumptionCN (5)

China’s rapid economic development has lead to an exponentially growing demand for energy, and China generates more than two-thirds of its energy by burning coal. During the period 1998 to 2008, China’s coal consumption increased at an average rate of 8% per year, and the 2008 consumption was about 2.1 billion tons of coal.

Use these data to predict China’s coal consumption in 2028.

If t = 0 represents 2008, Q0 = 2.1, r = 0.08, and t = 20 years.

Q = Q0 (1+r)t = 2.1 (1 + 0.08)20 = 2.1 (1.08)20 ≈ 9.8

China’s predicted coal consumption is about 9.8 billion tons.

a. Use the data to predict China’s coal consumption in 2020.

b. Make a graph projecting China’s coal consumption through 2050. Discuss the validity of the model.

9-C


Exponential growth functions have rates of change that increase.

Exponential decay functions have rates of change that decrease.

Linear functions have straight line graphs and constant rates of change. Exponential functions have graphs that rise or fall steeply and have variable rates of change.

Changing Rates of Change

9-CDrug ConcentrationCN (6a-b)

Consider an antibiotic that has a half-life in the bloodstream of 12 hours. A 10-milligraminjection of the antibiotic is given at 1:00 pm.

a. How much antibiotic remains int eh blood at 9:00 pm?

b. Draw a graph that shows the amount of antibiotic remaining as the drug is eliminated by the body.


9-CThe Allende MeteoriteCN (7)

The famous Allende meteorite lit up the skies of Mexico as it fell to Earth on Feb. 8, 1969. Scientists melted and chemically analyzed small pieces of the meteorite and found traces of both radioactive potatisium-40 and argon-40. From lab studies, we know that potassium-40 becomes argon-40 when it decays. Moreover, because argon-40 is a gas, it could not have been present when the meteorite first solidified. Scientists concluded that the argon-40 must have been trapped in the meteorite after it formed, in which case it must have come from decay of potassium-40. From the amounts of the two substances in the meteorite samples, the scientists determined that about 8.5% of the postatssium-40 originally present in the rock remains today. The half life of potassium-40 is about 1.25 billion years.

7. How old is the rock that makes up the Allende meteorite?


9-CQuick QuizCN (8)

8. Please answer the ten multiple choice questions from the Quick Quiz on p.546.


9-CHomework9C

9B Discussion Paragraph Class Notes 1-8 P.546:1-10 1 web

50. Radiometric Dating 51. Resource Consumption 52. Renewable Energy

1 world 53. Inflation Rate 54. Exponential Process 55. Radiometric Dating


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