Name: ________________________________________ Date __________________________ Period ______________
Population Growth, Earthquakes and Sound, pH Scale, Risk
I. Population Growth
The growth of the United States population can be approximately modeled by the
exponential formula y = P0ert where P0 is initial population t years ago and r is the growth
rate.
a. Visit the http://www.worldometers.info/world-population/us-population/
What is the current U.S. population? This will be your y-value _________________
Pick a population from another year? This will be your P0 value _________________
b. Use your answers to find the growth rate for the population of the United States between your chosen year
and currently in 2015. You need to know your time. Try to be as exact as you can. Write your answer as a
decimal.
c. Using your equation and your rate, what will the population be in 2020?
d. Scroll down on the website to find the forecasted table. How does your answer compare? What are some
reasons for any differences?
e. Using your equation and your rate, find the time at which the US population will reach 400,000,000
people. (Round the nearest year). Compare your answer to the forecasted table. How does it compare? What
are some reasons for any differences?
SOURCES: www.math.wsu.edu/faculty/cjacobs/105/105Worksheet2.doc
ww2.nwic.edu/tenrm/spring2001/Richter.htm
http://www.greenleecds.com/mathdownloads.html
II. Richter Scale:
Watch this video on the Richter scale before you begin: https://www.youtube.com/watch?v=1qbg7orb1lc
The Richter Scale is a logarithmic function that uses the common base, 10. The equation is: R = log I where R is the
strength of the earthquake on the Richter scale and I is the intensity of the quake
Now, given two Richter values you will be able to find how much greater in intensity one earthquake is as compared to
the other using the information below. Read below and then answer the following questions.
If earthquake #1 has a value of R = 5 and earthquake #2 has a value R = 8, how much greater in
intensity was the second earthquake compared to the first earthquake?
Method: Use the formula for logarithms: R = log I for each. Change the logarithmic
function to an exponential function and divide the two.
So, to compare log (I) = 8 to log (I) = 5, we need to change it to 108 and 105. Then we can use properties of
exponents to solve 108/105 = 108-5 = 103 =1,000. Meaning the second earthquake was 1000 times more intense.
1. Visit the "Rapid Earthquake Viewer” at: http://rev.seis.sc.edu/ . There is a lot of interesting information at this site, so
I hope you'll look around at it. We're just going to use it to get some earthquake magnitudes.
a. Click on the "Earthquake View" button on the viewer to select two earthquakes from the dropdown menu at
the bottom of the page. Where did they occur, and what was the magnitude of each?
1. __________________________________________________________
2. __________________________________________________________
b. Calculate the intensity of both earthquakes to find out how many times more intense was the stronger one.
c. Two 2010 Chile earthquake measured an 8.8 on the Richter scale, whereas the 2010 Haiti earthquake was a 7.0. How
many times more intense was the one in Chile?
d. Historically, it was known that the earthquake in Haiti was much more devastating. Why might this be? Do some
research if you are unsure.
III. Decibels
Another application of logarithmic functions is the way the human ear responds to
changes in sound intensity. You might be aware that the sound level of a stereo is
measured in decibels. (Did you know that the decibel was named after Alexander
Graham Bell?)
To figure out how many more times intense different sounds are, take the difference of
the two decibel sounds, then do 10decibel difference. For example, a jet engine has a dB of
140 and a normal conversation is at 60db. So, 140-60 = 80. So, the sounds are 108 or 100,000,000 times more
intense than each other.
a. There are many decibel charts online. Write down two sounds and their corresponding dB scale. Here is
one chart you can use: http://www.gcaudio.com/resources/howtos/loudness.html.
1. __________________________________________________________
2. __________________________________________________________
b. How many times louder (times more intense) is the rock concert than a normal conversation?
Go to the iPad decibel app. Hold it up to see the decibel that our room reaches. Make a louder noise (tap the
desk, stop your feet, yell-within reason, etc.) and write down the decibel.
a. Decibel read in room ____________ b. Decibel read when louder ___________
How many times louder was the second noise than the first?
IV. pH scale.
pH scale is also logarithmic. You may
have studied this in chemistry. Doctors
and dieticians also use the pH scale to
address health issues. Use the equation
below and try the following examples.
pH= − log [H+ ] where [H+ ] is the
concentration of the H+ ion.
Watch this video for a bit more on
the pH scale:
https://www.youtube.com/watch?v=3U9n4BV2618
1. James, an ulcer patient, has been told to avoid acidic foods. If he drinks coffee, which has a pH of 5.0 it
bothers him. His tap water with a pH of 5.8 is ok, as is milk, which has a pH of 6.9. Find the [H+]
concentration in the coffee, water and milk.
2. Find the [H+] concentration of a mixture which is half milk and half coffee. What is the pH of this
mixture? Why is this not just the average of the two pH’s?
3. If the H+ concentration of soda is about .003, what is the pH level of soda? Should James drink soda?
Why or why not?
4. Go online and find some acid and base pH scales. Find one food or drink he can have and one food or
drink he should have.
This is also an interesting video if you have time: http://ed.ted.com/lessons/steve-kelly-logarithms-explained
V. Risk
Actuaries are people who study and calculate risks. These professionals are very well paid and well regarded in
the business world. If you decide to study math, statistics, or financial majors in college, think about a job as
an actuary! Read below to understand more about risk and then answer the questions.
Every action is risky to some extent. Most people would consider some activities, such as hang gliding or
mountain climbing, quite risky. There are other things that are not commonly thought of as risky, such as
driving to school or taking a shower, yet even these carry some risk. The probability of various risks has been
calculated, but most of the numbers are so small that many people do not have an intuitive understanding of
them. To help better understand these types of probabilities, John and Sean Paling in their book Up to your
Armpits in Alligators? How to Sort Out What Risks Are Worth Worrying About, devised a logarithmic scale to
describe risks. This scale uses common
logarithms to convert a probability to a risk
number. Note that a one in a million chance is
assigned a risk number of 0. Events associated
with risk numbers below zero are insignificant
and should not be of much concern.
An asteroid colliding with earth: +.6 risk
number. Injuring yourself and needing
immediate medical attention: +5.5 risk number
1. The function which expresses risk number r in terms of probability p is r = log p +6. Show that this
equation works by plugging in a probability (p) to solve for the risk number (r). Then, plug in a risk
number (r) and solve for the probability (p).
2. An asteroid will collide with the earth has a risk number of +0.6. What is the probability of this happening?
3. They say that lightning never strikes twice in the same place. However, it has been known to strike
some people more than once in their lifetime.
a. The probability that lightning will strike you in your lifetime is 1 in 9000. What is the risk
number for being struck by lightning once?
b. Challenge: What is the risk number for being struck by lightning twice? Hint: First find the
probability that it will strike you twice.
4. Google about actuaries, see what you find, and write down a couple things you learned.
Source: http://academic.evergreen.edu