Exponential Growth and Decay
|
|
|
- Martha McCarthy
- 6 years ago
- Views:
Transcription
1 Exponential Growth and Decay 2 May 2012 Exponential Growth and Decay 2 May /24
2 On Monday we discussed population growth, and saw that it behaved like compound interest in how it grows. In particular, we saw that population growing at a constant rate r satisfies the equation P = P 0 e rt for the population t years after some initial time, where P 0 is the population at that time. Exponential Growth and Decay 2 May /24
3 On Monday we discussed population growth, and saw that it behaved like compound interest in how it grows. In particular, we saw that population growing at a constant rate r satisfies the equation P = P 0 e rt for the population t years after some initial time, where P 0 is the population at that time. The number e is called the base of the natural logarithm. Its value is approximately e = 2.7. Exponential Growth and Decay 2 May /24
4 Today we ll discuss another example, radioactive decay. The mathematics is similar, but with one difference. While money in your savings account grows, and population (generally) grows over time, the amount of a radioactive substance decreases over time. Exponential Growth and Decay 2 May /24
5 Today we ll discuss another example, radioactive decay. The mathematics is similar, but with one difference. While money in your savings account grows, and population (generally) grows over time, the amount of a radioactive substance decreases over time. We ll also see how exponential growth and decay applies to Google maps. Exponential Growth and Decay 2 May /24
6 Radioactive Decay Suppose you have a radioactive substance, such as Plutonium or Uranium. It decays into other substances. The amount that decays in a given time is proportional to the amount of the substance. The amount is then governed by the equation P 0 e rt where P 0 is the amount at a given time, r is the decay rate, and t is the amount of time past the initial time. Because the amount is decreasing rather than increasing, the minus sign shows up in the formula. Exponential Growth and Decay 2 May /24
7 Half Life The value of the decay rate tends to be a very small number, and not so convenient to remember. Instead, the amount of decay for a substance is usually determined by its half life. This is the length of time it takes for the substance to decay to half of its original amount. Exponential Growth and Decay 2 May /24
8 Half Life The value of the decay rate tends to be a very small number, and not so convenient to remember. Instead, the amount of decay for a substance is usually determined by its half life. This is the length of time it takes for the substance to decay to half of its original amount. Just like money gathering compound interest doubles in a fixed amount of time, regardless of how much is deposited, the amount of time it takes for a given substance to decay to half is the same regardless of how much of the substance you have. Exponential Growth and Decay 2 May /24
9 Clicker Question If a substance has a half life of 5,000 years, and you have an amount of it, after 10,000 years, how much of the original amount is remaining? A All B Half C A quarter D None E I don t know Exponential Growth and Decay 2 May /24
10 Answer After 5,000 years the amount reduces to half. After another 5,000 years the amount reduces to half of that, or half of a half. So, there is a quarter remaining. Exponential Growth and Decay 2 May /24
11 Answer After 5,000 years the amount reduces to half. After another 5,000 years the amount reduces to half of that, or half of a half. So, there is a quarter remaining. For example, if we started with 1 lb. of the substance, after 5,000 years we d have 0.5 lbs. After another 5,000 years we d have half of that, or = 0.25 lbs., or a quarter of what we started with. Exponential Growth and Decay 2 May /24
12 The radioactive substance Uranium-238 has a half life of about 4.5 billion years. The corresponding decay rate r is r = which indicates why it is not so convenient to work with r. An unstable form of Carbon, Carbon-14, has a half life of about 5720 years. Exponential Growth and Decay 2 May /24
13 The radioactive substance Uranium-238 has a half life of about 4.5 billion years. The corresponding decay rate r is r = which indicates why it is not so convenient to work with r. An unstable form of Carbon, Carbon-14, has a half life of about 5720 years. Because of the long half life of many dangerous substances produced in nuclear reactors, storing radioactive material is a difficult problem. Exponential Growth and Decay 2 May /24
14 If T is the half life of a substance, then the decay rate is related to T by P 0 e rt = 1 2 P 0 Exponential Growth and Decay 2 May /24
15 If T is the half life of a substance, then the decay rate is related to T by P 0 e rt = 1 2 P 0 or e rt = 1 2 Exponential Growth and Decay 2 May /24
16 If T is the half life of a substance, then the decay rate is related to T by P 0 e rt = 1 2 P 0 or e rt = 1 2 Taking logarithms gives the equation rt = ln(1/2) So, we can find r by dividing by the half life T, getting r = ln(1/2) T = ln(2) T A property of logarithms allows us to simplify as we did in the last step. Exponential Growth and Decay 2 May /24
17 Half Life Formula To summarize, the decay rate r is related to the half life time by the formula r = ln(2) half life Exponential Growth and Decay 2 May /24
18 Carbon Dating In living things the ratio of ordinary Carbon (Carbon-12) to Carbon-14 is fixed. Upon death the Carbon-14 decays while ordinary Carbon does not. Exponential Growth and Decay 2 May /24
19 Carbon Dating In living things the ratio of ordinary Carbon (Carbon-12) to Carbon-14 is fixed. Upon death the Carbon-14 decays while ordinary Carbon does not. By measuring the amount of Carbon-12 to Carbon-14, the age of the object can be approximated. Exponential Growth and Decay 2 May /24
20 Determining the Age of the Dead Sea Scrolls In 1990 a tiny piece of paper taken from the Dead Sea scrolls, believed to date back to the first century, A.D., was found to have 79% of the original Carbon-14 content remaining. We can use this information to approximate the age of the scrolls. Exponential Growth and Decay 2 May /24
21 Determining the Age of the Dead Sea Scrolls In 1990 a tiny piece of paper taken from the Dead Sea scrolls, believed to date back to the first century, A.D., was found to have 79% of the original Carbon-14 content remaining. We can use this information to approximate the age of the scrolls. The scrolls were written on parchment and on papyrus. Parchment was made from animal skin and papyrus from the papyrus plant. In both cases it was made from a living substance. Carbon dating then dates how long the living thing has been dead. Exponential Growth and Decay 2 May /24
22 Determining the Age of the Dead Sea Scrolls In 1990 a tiny piece of paper taken from the Dead Sea scrolls, believed to date back to the first century, A.D., was found to have 79% of the original Carbon-14 content remaining. We can use this information to approximate the age of the scrolls. The scrolls were written on parchment and on papyrus. Parchment was made from animal skin and papyrus from the papyrus plant. In both cases it was made from a living substance. Carbon dating then dates how long the living thing has been dead. Since it is unlikely the animal skin was around for hundreds of years before being turned into parchment, or the papyrus harvested long before used, the length of time since death is a good approximation for when the scrolls were written. Exponential Growth and Decay 2 May /24
23 If r is the decay rate for Carbon-14, then to date the Dead Sea Scrolls we d like to solve for the time t satisfying P 0 e rt =.79 P 0 since t is the amount of time that allows a substance to have 79% of its Carbon-14 remaining, the case for the scrolls. If we divide by the (unknown) initial amount P 0 of Carbon-14, we get Exponential Growth and Decay 2 May /24
24 If r is the decay rate for Carbon-14, then to date the Dead Sea Scrolls we d like to solve for the time t satisfying P 0 e rt =.79 P 0 since t is the amount of time that allows a substance to have 79% of its Carbon-14 remaining, the case for the scrolls. If we divide by the (unknown) initial amount P 0 of Carbon-14, we get Taking logarithms, we get e rt =.79 t = ln(.79) r = 1945 This calculation indicates the scrolls were written around the 1st century, A.D. Exponential Growth and Decay 2 May /24
25 Another Example By measuring the Uranium-238 content of some moon rocks, it was found that there was 70% of the original amount of Uranium-238 remains today. We can use this to approximate the age of the rocks. Exponential Growth and Decay 2 May /24
26 Another Example By measuring the Uranium-238 content of some moon rocks, it was found that there was 70% of the original amount of Uranium-238 remains today. We can use this to approximate the age of the rocks. The only difference between this calculation and the previous one is in the value of the decay rate r and the percentage remaining. Here we have r = ln(2) 4.5 billion Exponential Growth and Decay 2 May /24
27 Another Example By measuring the Uranium-238 content of some moon rocks, it was found that there was 70% of the original amount of Uranium-238 remains today. We can use this to approximate the age of the rocks. The only difference between this calculation and the previous one is in the value of the decay rate r and the percentage remaining. Here we have r = ln(2) 4.5 billion The age t of the rocks we can approximate by the formula t = ln(0.70) r = ln(0.70) 4.5 billion ln(2) = 2.3 billion years Exponential Growth and Decay 2 May /24
28 Newton s Law of Cooling If you remove a loaf of bread from an oven, the temperature of the loaf will drop in a similar way to the way a radioactive substance decays. Newton s law of cooling (discovered by Isaac Newton), says that if T 0 is the initial temperature of the bread upon removing from the oven, and if T room is the room temperature, then after t hours, the temperature T is given by the formula Exponential Growth and Decay 2 May /24
29 Newton s Law of Cooling If you remove a loaf of bread from an oven, the temperature of the loaf will drop in a similar way to the way a radioactive substance decays. Newton s law of cooling (discovered by Isaac Newton), says that if T 0 is the initial temperature of the bread upon removing from the oven, and if T room is the room temperature, then after t hours, the temperature T is given by the formula T = T room + (T 0 T room ) e rt where r represents the cooling rate of the loaf of bread. Exponential Growth and Decay 2 May /24
30 Newton s Law of Cooling If you remove a loaf of bread from an oven, the temperature of the loaf will drop in a similar way to the way a radioactive substance decays. Newton s law of cooling (discovered by Isaac Newton), says that if T 0 is the initial temperature of the bread upon removing from the oven, and if T room is the room temperature, then after t hours, the temperature T is given by the formula T = T room + (T 0 T room ) e rt where r represents the cooling rate of the loaf of bread. The rate depends on factors such as density and composition of the loaf of bread. In order to be able to calculate with this formula, we need to know the room temperature, and the temperature of the bread at two different times. Exponential Growth and Decay 2 May /24
31 Time of Death Determination A primary application of Newton s law of cooling is determining time of death. By measuring the temperature of the body at two different times, then the time of death can be approximated. Exponential Growth and Decay 2 May /24
32 Time of Death Determination A primary application of Newton s law of cooling is determining time of death. By measuring the temperature of the body at two different times, then the time of death can be approximated. For example, suppose the temperature of a dead body is measured at 8:00 a.m. and at 9:00 a.m. The temperatures are 97.1 and 96.8, respectively. If the room where the body lies is around 65, approximately how long ago did the person die? Exponential Growth and Decay 2 May /24
33 We can set our initial time to be 8:00 a.m., so our initial temperature is Our room temperature is 65. Our equation simplifies a little to T = 65 + ( ) e rt = e rt Exponential Growth and Decay 2 May /24
34 When t = 1, we are at 9:00 a.m., and T = So, or 96.8 = e r 31.8 = 32.1 e r This allows us to calculate r, getting ( ) 31.8 r = ln = Exponential Growth and Decay 2 May /24
35 A typical assumption is that, when a person dies, their body temperature is We then want to find the time where T = In other words, we want to find t where 98.6 = e rt Going through the algebra will give t = 4.9. The negative value means the time of death is before our initial time, which makes sense. In other words, the person died about 5 hours before 8:00 a.m., or around 3:00 a.m. Exponential Growth and Decay 2 May /24
36 Google Zooming Why can you start with a Google map of the earth, or a county, and zoom in on a city or neighborhood so quickly? Let s investigate this. Exponential Growth and Decay 2 May /24
37 Google Zooming Why can you start with a Google map of the earth, or a county, and zoom in on a city or neighborhood so quickly? Let s investigate this. We ll look at the scale indicator on a google map. Going to maps.google.com will give a map of the U.S. The scale is approximately 1 inch equals 500 miles (at least on my computer). Exponential Growth and Decay 2 May /24
38 Google Zooming Why can you start with a Google map of the earth, or a county, and zoom in on a city or neighborhood so quickly? Let s investigate this. We ll look at the scale indicator on a google map. Going to maps.google.com will give a map of the U.S. The scale is approximately 1 inch equals 500 miles (at least on my computer). We can zoom in or out by using the zoom bar, or clicking on the + or signs at the top and bottom, respectively, of the zoom bar. Exponential Growth and Decay 2 May /24
39 Google Zooming Why can you start with a Google map of the earth, or a county, and zoom in on a city or neighborhood so quickly? Let s investigate this. We ll look at the scale indicator on a google map. Going to maps.google.com will give a map of the U.S. The scale is approximately 1 inch equals 500 miles (at least on my computer). We can zoom in or out by using the zoom bar, or clicking on the + or signs at the top and bottom, respectively, of the zoom bar. Each click on the Google + sign for zooming approximately halves the scale. That is, if one inch represents 500 miles, clicking on the plus sign once changes the scale to 1 inch represents 250 miles. Exponential Growth and Decay 2 May /24
40 A Question If you have a map of the U.S., how many clicks do you need for NMSU to take up most of the screen? Exponential Growth and Decay 2 May /24
41 A Question If you have a map of the U.S., how many clicks do you need for NMSU to take up most of the screen? The U.S. is about 3,000 miles wide. The distance across NMSU, from El Paso to I25 along Stewart Avenue is about 1.5 miles. The distance across the U.S. is then about the distance across NMSU = 2000 times Exponential Growth and Decay 2 May /24
42 The number of clicks on + or on the Google map amounts to the number of times we double or half the map scale. To change the scale by a factor of 2000, we effectively are asking for which n is 2 n = By using logarithms, we ll get which is about 11. n = ln(2000) ln(2) Exponential Growth and Decay 2 May /24
43 The number of clicks on + or on the Google map amounts to the number of times we double or half the map scale. To change the scale by a factor of 2000, we effectively are asking for which n is 2 n = By using logarithms, we ll get which is about 11. n = ln(2000) ln(2) So, it takes 11 clicks of the Google + to magnify to have NMSU show up as big as the U.S. looked originally. Exponential Growth and Decay 2 May /24
44 Scale Here is a two-minute YouTube video showing the effect of exponential growth by relating it to sizes of astronomical objects. Video of sizes of planets, stars, and galaxies. Exponential Growth and Decay 2 May /24
45 Scale Here is a two-minute YouTube video showing the effect of exponential growth by relating it to sizes of astronomical objects. Video of sizes of planets, stars, and galaxies. Here is a website that shows both very large and very small objects, reflecting how quickly sizes can change which grow exponentially. htwins.net/scale2 Exponential Growth and Decay 2 May /24
46 Next Time We ll end the semester by identifying some mathematical and scientific ideas that show up in an episode of the Simpsons. During the episode Treehouse of Horror VI, one of the segment involves Homer getting trapped in the third dimension. We ll discuss briefly the ideas and then watch the relevant segment to see how the writers brought in those ideas. Exponential Growth and Decay 2 May /24
Chapter 8: Exponential Astonishment Lecture notes Math 1030 Section B
Section B.1: Doubling Time Definition of doubling time The time required for each doubling in exponential growth is called the doubling time. After a time t, an exponentially growing quantity with a doubling
7.5. Exponential Growth and Decay. 502 Chapter 7: Transcendental Functions. The Law of Exponential Change
502 Chapter 7: Transcendental Functions 7.5 Exponential Growth and Decay Exponential functions increase or decrease very rapidly with changes in the independent variable. They describe growth or decay
a)! War b)! Poisoning c)! Melanoma (skin cancer) d)! STDs (not counting HIV/AIDS) e)! Astronomy related deaths
Leslie Looney This Class (Lecture 3): Phone: 244-3615 Astro-Death is very unlikely Email: lwl1@1uiuc1.1edu Office: Astro Building #218 Office Hours: Next Class: W: 11:00-11:59 a.m. Asteroids/Comets or
C. HECKMAN TEST 1A SOLUTIONS 170
C. HECKMAN TEST 1A SOLUTIONS 170 1) Thornley s Bank of Atlanta offers savings accounts which earn 4.5% per year. You have $00, which you want to invest. a) [10 points] If the bank compounds the interest
1. If (A + B)x 2A =3x +1forallx, whatarea and B? (Hint: if it s true for all x, thenthecoe cients have to match up, i.e. A + B =3and 2A =1.
Warm-up. If (A + B)x 2A =3x +forallx, whatarea and B? (Hint: if it s true for all x, thenthecoe cients have to match up, i.e. A + B =3and 2A =.) 2. Find numbers (maybe not integers) A and B which satisfy
KNOWLEDGE TO GET FROM TODAY S CLASS MEETING
KNOWLEDGE TO GET FROM TODAY S CLASS MEETING Class Meeting #5, Friday, January 29 th, 2016 1) GRAVITY: (text pages 111-112, 123) 2) Isaac Newton s LAWS of MOTION (briefly) (text pages 115-117) 3) Distances
Lesson 6 MA Nick Egbert
Overview In this lesson we start our stu of differential equations. We start by considering only exponential growth and decay, and in the next lesson we will extend this idea to the general method of separation
171S5.6o Applications and Models: Growth and Decay; and Compound Interest November 21, 2011
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
6.4 Exponential Growth and Decay
496 differential equations 6.4 Exponential Growth and Decay The separable differential equation y = y is relatively simple to solve, but it can model a wealth of important situations, including population
Section 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models
Section 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models 197 Objective #1: Find Equations of Populations that Obey the Law of Uninhibited Growth. In the last section, we saw that when interest
Introduction Growthequations Decay equations Forming differential equations Case studies Shifted equations Test INU0115/515 (MATHS 2)
GROWTH AND DECAY CALCULUS 12 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Growth and Decay 1/ 24 Adrian Jannetta Introduction Some of the simplest systems that can be modelled by differential
Section 2.3: Logarithmic Functions Lecture 3 MTH 124
Procedural Skills Learning Objectives 1. Build an exponential function using the correct compounding identifiers (annually, monthly, continuously etc...) 2. Manipulate exponents algebraically. e.g. Solving
Lifestyle Chemistry Intro
Lifestyle Chemistry Intro Year 10 Science - Topic One Lifestyle Chemistry What is this topic about? Often without realising it, you are exposed to chemistry in your everyday lives. This topic introduces
Practice Questions for Final Exam - Math 1060Q - Fall 2014
Practice Questions for Final Exam - Math 1060Q - Fall 01 Before anyone asks, the final exam is cumulative. It will consist of about 50% problems on exponential and logarithmic functions, 5% problems on
Chapter 13 - Inverse Functions
Chapter 13 - Inverse Functions In the second part of this book on Calculus, we shall be devoting our study to another type of function, the exponential function and its close relative the Sine function.
Algebra II. Slide 1 / 261. Slide 2 / 261. Slide 3 / 261. Linear, Exponential and Logarithmic Functions. Table of Contents
Slide 1 / 261 Algebra II Slide 2 / 261 Linear, Exponential and 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 261 Linear Functions Exponential Functions Properties
[2]
![[2]](/thumbs/79/78842361.jpg "[2]")
1 Fossil fuel power stations generate electricity. Nuclear power stations also generate electricity. (a) Many people think that nuclear power stations are a greater risk to people than fossil fuel power
Recursive Routines. line segments. Notice that as you move from left to right, the
CONDENSED LESSON 6. Recursive Routines In this lesson you will explore patterns involving repeated multiplication write recursive routines for situations involving repeated multiplication look at tables
Pre-AP Algebra 2 Unit 9 - Lesson 9 Using a logarithmic scale to model the distance between planets and the Sun.
Pre-AP Algebra 2 Unit 9 - Lesson 9 Using a logarithmic scale to model the distance between planets and the Sun. Objectives: Students will be able to read a graph with a logarithmic scale. Students will
History of the Universe Unit Tracking Sheet
Name Period Mrs. Coates Earth Science History of the Universe Unit Tracking Sheet Learning Target Question Example Date Target was Taught in Class The Big Bang Theory explains how the universe formed The
Read It! Station Directions
Read It! Station Directions Each member of the group will read the passage and answer the questions from the task cards on the lab sheet in the Read It! section. It is important to remember that the answers
c) O-16 d) Pu An unstable nucleus emits. a) Atoms b) Electricity c) Plasma d) Radiation 3. Many of uranium are radioactive. a) Ions b) Isomers
Physical Science Domain 1 Nuclear Decay Review 1. Which nucleus would be MOST likely to be radioactive? a) C-12 b) Ca-40 c) O-16 d) Pu-241 2. An unstable nucleus emits. a) Atoms b) Electricity 3. Many
Half Life Introduction
Name: Date: Period: Half Life Introduction The half-life of an element is the time it will take half of the parent atoms to transmutate into different atoms (through alpha or beta decays, or another process).
NUMB3RS Activity: How to Get a Date. Episode: Bones of Contention
Teacher Page NUMB3RS Activity: How to Get a Date Topic: Exponential Decay Grade Level: 0 - Objective: To learn about exponential decay and its application to radiocarbon dating. Time: About 30 minutes
(a) If the half-life of carbon-14 is 5,730 years write the continuous growth formula.
Section 6.7: Exponential and Logarithmic Models In this text all application problems are going to be of the following form, where A 0 is the initial value, k is the growth/decay rate (if k > 0 it is growth,
TABLE OF CONTENTS. click one to go to that page, or just go on. What is the Solar System? Neptune (Pluto) The Sun. Asteroids. Mercury.
The Solar System TABLE OF CONTENTS click one to go to that page, or just go on. What is the Solar System? The Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune (Pluto) Asteroids Meteors and Meteorites
What is the solar system?
Notes Astronomy What is the solar system? 11.1 Structure of the Solar System Our solar system includes planets and dwarf planets, their moons, a star called the Sun, asteroids and comets. Planets, dwarf
o Terms to know o Big Bang Theory o Doppler Effect o Redshift o Universe
Standard 1: Students will understand the scientific evidence that supports theories that explain how the universe and the solar system developed. They will compare Earth to other objects in the solar system.
Aim:How can we determine the particles emitted from radioactive
Aim:How can we determine the particles emitted from radioactive decay? Nuclear Stability: The larger (more massive) a nucleus is, the harder it is for it to stay together. Natural transmutation (natural
MATH 1101 Exam 3 Review - Gentry. Spring 2018
MATH 1101 Exam 3 Review - Gentry Spring 2018 Topics Covered Section 5.3 Fitting Exponential Functions to Data Section 5.4 Logarithmic Functions Section 5.5 Modeling with Logarithmic Functions What s in
Exponential and Logarithmic Equations
OpenStax-CNX module: m49366 1 Exponential and Logarithmic Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section,
Thinking Like a Chemist About Nuclear Change!
Thinking Like a Chemist About Nuclear Change! What are we going to learn today? Nuclear Changes REACTIONS ENERGY RELEASED DECAY Poll: Clicker Question There was a nuclear emergency in Japan. The emergency
Our Place in the Universe (Chapter 1) The Structure and Size of the Universe
Our Place in the Universe (Chapter 1) The Structure and Size of the Universe Based on Chapter 1 This material will be useful for understanding Chapters 2, 3, and 13 on Years, Seasons, and Months, The Orbits
nuclear chemical change CH4 + 2O2 CO2 + 2H2O carbon dating
Nuclear Chemistry I. What is nuclear chemistry? a. Nuclear changes vs. chemical changes i. A nuclear change is a change in which the nucleons (things in the nucleus) change. For instance, if the number
You Might Also Like. I look forward helping you focus your instruction while saving tons of time. Kesler Science Station Lab Activities 40%+ Savings!
Thanks Connect Thank you for downloading my product. I truly appreciate your support and look forward to hearing your feedback. You can connect with me and find many free activities and strategies over
Unit 2 Lesson 3 Absolute Dating. Copyright Houghton Mifflin Harcourt Publishing Company
It s About Time! How can the absolute age of rock be determined? Determining the actual age of an event or object in years is called absolute dating. Scientists often use radioactive isotopes to find the
Earth & Space. Learning Target:
Earth & Space Learning Target: Most of the cycles and patterns of motion between the Earth and sun are predictable. You understand this when you know: 1) 1 Revolution of the Earth takes approximately 365
Where, on Earth, are you?
Where, on Earth, are you? We all live on a big planet called Earth, of course. But where on Earth? And if you want to go to a different spot on Earth other than the spot you are in, how do you know which
Solving Exponential Equations (Applied Problems) Class Work
Solving Exponential Equations (Applied Problems) Class Work Objective: You will be able to solve problems involving exponential situations. Quick Review: Solve each equation for the variable. A. 2 = 4e
MATH 236 ELAC FALL 2017 CA 10 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MATH 36 ELAC FALL 7 CA MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) In a certain country, the rate of increase of the population is proportional
Mathematics and the Simpsons. Mathematics and the Simpsons 1/13
Mathematics and the Simpsons Mathematics and the Simpsons 1/13 This semester we ve occasionally used the Simpsons to introduce ideas and to help explain them. That show has lots of math and science references
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.8 Exponential Growth and Decay In this section, we will: Use differentiation to solve real-life problems involving exponentially growing quantities. EXPONENTIAL
5.6 Applications and Models: Exponential Growth and Decay
5.6 Applications and Models: Exponential Growth and Decay Many natural (and business/financial) processes build on themselves exponentially. We will see several examples. All of these examples are functions
Properties of Logarithms. Example Expand the following: The Power Rule for Exponents - (b m ) n = b mn. Example Expand the following: b) ln x
Properties of Logarithms The Product Rule for Exponents - b m b n = b m+n Example Expand the following: a) log 4 (7 5) log b MN = log b M + log b N b) log (10x) The Power Rule for Exponents - (b m ) n
Sec. 4.2 Logarithmic Functions
Sec. 4.2 Logarithmic Functions The Logarithmic Function with Base a has domain all positive real numbers and is defined by Where and is the inverse function of So and Logarithms are inverses of Exponential
An equation of the form y = ab x where a 0 and the base b is a positive. x-axis (equation: y = 0) set of all real numbers
Algebra 2 Notes Section 7.1: Graph Exponential Growth Functions Objective(s): To graph and use exponential growth functions. Vocabulary: I. Exponential Function: An equation of the form y = ab x where
Absolute Dating. Using half-lives to study past-lives. Notes #26
Absolute Dating Using half-lives to study past-lives Notes #26 What is radioactivity? (notes) Some atoms have an unstable nucleus Over time, these nuclei* fall apart, creating two smaller atoms (radioactive
MATH 1101 Exam 3 Review. Spring 2018
MATH 1101 Exam 3 Review Spring 2018 Topics Covered Section 5.3 Fitting Exponential Functions to Data Section 5.4 Logarithmic Functions Section 5.5 Modeling with Logarithmic Functions Section 4.1 Systems
LESSON 1. Solar System
Astronomy Notes LESSON 1 Solar System 11.1 Structure of the Solar System axis of rotation period of rotation period of revolution ellipse astronomical unit What is the solar system? 11.1 Structure of the
Today: 5.4 General log and exp functions (continued) Warm up:
Today: 5.4 General log and exp functions (continued) Warm up: log a (x) =ln(x)/ ln(a) d dx log a(x) = 1 ln(a)x 1. Evaluate the following functions. log 5 (25) log 7 p 7 log4 8 log 4 2 2. Di erentiate the
Q1. Describe, in as much detail as you can, the life history of a star like our Sun
Q1. Describe, in as much detail as you can, the life history of a star like our Sun..................................... (Total 6 marks) Q2. The energy radiated by a main sequence star like the Sun is
4.4 Atomic structure Notes
4.4 Atomic structure Notes Ionising radiation is hazardous but can be very useful. Although radioactivity was discovered over a century ago, it took many nuclear physicists several decades to understand
3.8 Exponential Growth and Decay
October 15, 2010 Population growth Population growth If y = f (t) is the number of individuals in a population of animals or humans at time t, then it seems reasonable to expect that the rate of growth
Big Bang, Black Holes, No Math
ASTR/PHYS 109 Dr. David Toback Lecture 6 & 7 1 Prep For Today (is now due) L7 Reading: Required: BBBHNM Unit 2 (Chapters 5-9) Recommended Reading: See P3 of http://people.physics.tamu.edu/toback/109/syllabus.pdf
Assignment #0 Using Stellarium
Name: Class: Date: Assignment #0 Using Stellarium The purpose of this exercise is to familiarize yourself with the Stellarium program and its many capabilities and features. Stellarium is a visually beautiful
Section 6.3. x m+1 x m = i n x m, (6.3.1)
Difference Equations to Differential Equations Section 6.3 Models of Growth and Decay In this section we will look at several applications of the exponential and logarithm functions to problems involving
Alpha Decay Simulation
Alpha Decay Simulation Go to class website, select Computer Simulations, Physics 30, Alpha Decay. Part A: Radioactive Decay in a single atom: go to the SECOND tab (single atom). 1. In the simulation, a
SHORT DISCOVERY-BASED STEM EXPERIENCES STEM. Brought to you by the NATIONAL AFTERSCHOOL ASSOCIATION
SHORT DISCOVERY-BASED STEM EXPERIENCES STEM gems Brought to you by the NATIONAL AFTERSCHOOL ASSOCIATION SOLAR ECLIPSE big IDEAS ON MONDAY AUGUST 21, 2017 NORTH AMERICA WILL BE TREATED TO A RARE CELESTIAL
Inner Transition Metals
1 Inner Transition Metals Inner Transition Metals Inner Transition Metals The inner transition metals are found in the f-block, usually put at the bottom of the Periodic Table. These elements were sometimes
What we will learn about Fossils?
The fossil Record What we will learn about Fossils? We will learn how fossils form? What types of fossils exist? How do scientists determine the age of fossils? What do fossils reveal to scientists? What
Thinking Like a Chemist About Nuclear Change!
UNIT7-DAY-LaB230 Page UNIT7-DAY-LaB230 Thursday, March 07, 203 8:48 AM Thinking Like a Chemist About Nuclear Change! IMPORTANT INFORMATION Begin Unit 7 LM s assigned on March 9 th HW assigned on March
Assignment #12 The Milky Way
Name Date Class Assignment #12 The Milky Way For thousands of years people assumed that the stars they saw at night were the entire universe. Even after telescopes had been invented, the concept of a galaxy
Earth s History. The principle of states that geologic processes that happened in the past can be explained by current geologic processes.
Earth s History Date: Been There, Done That What is the principle of uniformitarianism? The principle of states that geologic processes that happened in the past can be explained by current geologic processes.
Section 4.6 Notes. Modeling with Exponential and Logarithmic Equations
Section 4.6 Notes Modeling with Exponential and Logarithmic Equations Exponential Growth A population that experiences exponential growth will increase according to the equation n(t) is the population
Exponential Growth and Decay
Exponential Growth and Decay 2-4-2005 In certain situations, the rate at which a thing grows or decreases is proportional to the amount present. When a substance undergoes radioactive decay, the release
Math 180 Chapter 4 Lecture Notes. Professor Miguel Ornelas
Math 80 Chapter 4 Lecture Notes Professor Miguel Ornelas M. Ornelas Math 80 Lecture Notes Section 4. Section 4. Inverse Functions Definition of One-to-One Function A function f with domain D and range
Understanding the Universe S TA R T ING WITH EARTH A ND B E YO ND
Unit Overview: Understanding the Universe S TA R T ING WITH EARTH A ND B E YO ND Our solar system examining size and scale in space 6.11B UNDERSTAND THAT GRAVITY IS THE FORCE THAT GOVERNS MOTION IN OUR
You Might Also Like. I look forward helping you focus your instruction while saving tons of time. Kesler Science Station Lab Activities 40%+ Savings!
Thanks Thank you for downloading my product. I truly appreciate your support and look forward to hearing your feedback. Connect You can connect with me and find many free activities and strategies over
Telescopes and Observatories. Parent Guide, page 1 of 2. OurStory: Exploring the Sky. Read the Directions sheets for step-by-step instructions.
Parent Guide, page 1 of 2 Read the Directions sheets for step-by-step instructions. SUMMARY In this activity, children and adults will watch and discuss a short video about how people use telescopes. WHY
Q1. What does the word biology mean? Q4. If you stacked 4.6 billion pieces of paper on top of each other, how high would the pile be?
This website is designed to help you gain a better understanding of biology. You need to complete the series of activities outlined below and answer any questions on this sheet. Have fun! Go to the website:
Do you know how to find the distance between two points?
Some notation to understand: is the line through points A and B is the ray starting at point A and extending (infinitely) through B is the line segment connecting points A and B is the length of the line
Intermediate Algebra. 8.6 Exponential Equations and Change of Base. Name. Problem Set 8.6 Solutions to Every Odd-Numbered Problem.
8. Exponential Equations and Change of Base 1. Solving the equation: 3. Solving the equation: 3 x = 5 5 x = 3 x = ln5 x = ln5 ln5 x = x ln5 = x = ln5 1.450 x = ln5 0.82 5. Solving the equation: 7. Solving
Growth 23%
y 100 0. 4 x Decay 23% Math 109C - Fall 2012 page 16 39. Write the quantity 12,600,000,000 miles in scientific notation. The result is: (A) 12. 6 x 10 9 miles (B) 12. 6 x 10 9 miles (C) 1. 26 x 10 10 miles
Exponential Growth (Doubling Time)
Exponential Growth (Doubling Time) 4 Exponential Growth (Doubling Time) Suppose we start with a single bacterium, which divides every hour. After one hour we have 2 bacteria, after two hours we have 2
#2 Points possible: 1. Total attempts: 2 An exponential function passes through the points (0, 3) and (3, 375). What are the
Week 9 Problems Name: Neal Nelson Show Scored View #1 Points possible: 1. Total attempts: 2 For each table below, could the table represent a function that is linear, exponential, or neither? f(x) 90 81
33. The gas law for an ideal gas at absolute temperature T (in. 34. In a fish farm, a population of fish is introduced into a pond
SECTION 3.8 EXPONENTIAL GROWTH AND DECAY 2 3 3 29. The cost, in dollars, of producing x yards of a certain fabric is Cx 1200 12x 0.1x 2 0.0005x 3 (a) Find the marginal cost function. (b) Find C200 and
Class: 6 Science Date: 1/30/13 Solar System Simulator + Guide
Class: 6 Science Date: 1/30/13 Solar System Simulator + Guide Objectives Today we will learn information about each planet and continue to familiarize ourselves with the solar system. We will learn this
R A D I A T I O N P R O T E C T I O N a n d t h e N R C
R A D I A T I O N P R O T E C T I O N and the NRC Radiation is all around us. It is naturally present in our environment and has been since before the birth of this planet. Radiation occurs in nature,
The Mass of Jupiter Student Guide
The Mass of Jupiter Student Guide Introduction: In this lab, you will use astronomical observations of Jupiter and its satellites to measure the mass of Jupiter. We will use the program Stellarium to simulate
Section 5.6. Applications and Models: Growth and Decay; Compound
Section 5.6 Applications and Models: Growth and Decay; Compound Interest Exponential Growth A quantity that experiences exponential growth will increase according to the equation P(t) = P 0 e kt t is the
Study Guide and Review - Chapter 7
Choose a word or term from the list above that best completes each statement or phrase. 1. A function of the form f (x) = b x where b > 1 is a(n) function. exponential growth 2. In x = b y, the variable
NUCLEAR CHEMISTRY. Physical Science Spring 2017 NAME: CLASS PERIOD: TEACHER: PAGE NUMBERS. LAB POINTS EARNED Guided Notes: Types of Radiation 2
NUCLEAR CHEMISTRY Physical Science Spring 2017 NAME: CLASS PERIOD: TEACHER: ASSIGNMENT PAGE NUMBERS DUE DATE HW POINTS EARNED LAB POINTS EARNED Guided Notes: Types of Radiation 2 Learning Targets: Nuclear
Cool Video Clip Image Source: Google
Cool Video Clip 1. 2. 1 2 The theme for Seventh Grade Science is structure. Structure is how things are built. It often refers to the physical characteristics of something. 3 All substances are made of
What s the longest single-shot exposure ever recorded of any object or area of space by Hubble?
Hubblecast Episode 50: Q&A with Dr J 00:00 Have you ever wondered why Hubble can make detailed images of of galaxies, but stars appear as featureless blobs? What the most distant object ever observed is?
DISCUSS DISCOVER PROVE WRITE. (a) log a x y b log x log y (b) log 2 1x y2 log 2 x log 2 y. (c) log 5 a a b 2 b log 5 a 2 log 5 b
360 CHAPTER 4 Exponential and Logarithmic Functions 74. Biodiversity Some biologists model the number of species S in a fixed area A (such as an island) by the species-area relationship log S log c k log
Solar System Unit Tracking Sheet
Name Period Mrs. Coates Earth Science Solar System Unit Tracking Sheet Learning Target The Solar System is 4.6 billion years old Question Example How is the age of the solar system estimated by scientists?
CHARTING THE HEAVENS USING A VIRTUAL PLANETARIUM
Name Partner(s) Section Date CHARTING THE HEAVENS USING A VIRTUAL PLANETARIUM You have had the opportunity to look at two different tools to display the night sky, the celestial sphere and the star chart.
Chapter 4.2: Exponential & Logistic Modeling
Chapter 4.2: Exponential & Logistic Modeling For real-life applications, our independent variable is usually time, t. Example 1: Start with an initial value of $100. Assuming this amount increases by 30%
Leader Discussion Guide for Cosmos: A SpaceTime Odyssey
Leader Discussion Guide for Cosmos: A SpaceTime Odyssey Episode 1: Standing Up in the Milky Way The creators of Cosmos: A SpaceTime Odyssey state that their aim is to promote scientific literacy. We know
Introduction to the Night Sky
Slide 1 / 106 Slide 2 / 106 New Jersey Center for Teaching and Learning Progressive Science Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use
MATH 1113 Exam 2 Review. Spring 2018
MATH 1113 Exam 2 Review Spring 2018 Section 3.1: Inverse Functions Topics Covered Section 3.2: Exponential Functions Section 3.3: Logarithmic Functions Section 3.4: Properties of Logarithms Section 3.5:
Section 4.5. Using Exponential Functions to Model Data
Section 4.5 Using Exponential Functions to Model Data Exponential Model, Exponential Related, Approximately Exponentially Related Using Base Multiplier Property to Find a Model Definition An exponential
! What the hell happened to Pluto?! The speed of light!! Scale of the Solar System
Leslie Looney This Class (Lecture 2): Phone: 244-3615 Email: lwl1@1illinois1.1edu Poor Pluto Office: Astro Building #218 Office Hours: Next Class: R: 10:00-11:00 a.m. Astro-Death is very unlikely or by
ASTR 380. The Universe: the context for Life
ASTR 380 The Universe: the context for Life Simple facts: The Universe is vast. The Universe is old. The elements for life are wide-spread. Our physical laws appear universal The Universe is mostly empty!
Exponents and Logarithms
chapter 5 Starry Night, painted by Vincent Van Gogh in 1889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Exponents and Logarithms This chapter focuses on understanding
SOLAR SYSTEM, STABILITY OF ORBITAL MOTIONS, SATELLITES
SOLAR SYSTEM, STABILITY OF ORBITAL MOTIONS, SATELLITES Q1. The figure below shows what scientists over 1000 years ago thought the solar system was like. Give one way that the historical model of the solar
How old is Earth? Bishop of Ireland James Ussher
Geologic Time How old is Earth? Bishop of Ireland James Ussher Not so long ago, the Bible was regarded as the ultimate source of truth. Through the 18th and 19th centuries it was believed that the world
ACTIVITIES for Grades 9-12
ACTIVITIES for Grades 9-12 BEFORE YOUR VISIT Online Video: Journey to the Stars Trailer and Prelude To prepare for your Museum visit, watch the trailer and the prelude with your students. Class Discussion:
Solve Exponential and Logarithmic Equations. You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations.
TEKS 7.6 Solve Exponential and Logarithmic Equations 2A..A, 2A..C, 2A..D, 2A..F Before Now You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations. Why?
Outer space: A matter of gravity
1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,