home


 * Home Work Pictures**








 * Ryan Sullivan and Arthur Xu**

__**Earthquakes and Richter Magnitude**__

media type="youtube" key="vqOTJnJc8Is" height="315" width="420"

On Tuesday, August 23, 2011, two sizable earthquakes occurred in the US. One was in Colorado with a magnitude of 5.3 and the other was centered in Virginia with a magnitude of 5.8 … which we felt here in Massachusetts. Seismologists view images on seismographs like this image on the right from Tuesday’s earthquake. They calculate the vertical distance between the extremes on this image and call that the amplitude of the shake. The Richter Magnitude of the earthquake is the base-10 logarithm of that amplitude. Logarithms are related to exponents. Base-10 logarithms are the exponent required to bring 10 to a certain number.

Let's take a look at the seismic wave energy yielded by our two recent U.S. examples of recent activity and compare those to earthquakes and other phenomena. For this we'll use a larger unit of energy, the seismic energy yield of quantities of the explosive TNT:


 * Richter TNT for Seismic Example**


 * Magnitude Energy Yield (approximate)**


 * -1.5**- 6 ounces Breaking a rock on a lab table


 * 1.0**- 30 pounds Large Blast at a Construction Site


 * 1.5**- 320 pounds


 * 2.0**- 1 ton Large Quarry or Mine Blast


 * 2.5**- 4.6 tons


 * 3.0**- 29 tons


 * 3.5**- 73 tons


 * 4.0-** 1,000 tons Small Nuclear Weapon


 * 4.5**- 5,100 tons Average Tornado (total energy)


 * 5.0**- 32,000 tons


 * 5.5**- 80,000 tons Little Skull Mtn., NV Quake, 1992


 * 6.0**- 1 million tons Double Spring Flat, NV Quake, 1994


 * 6.5**- 5 million tons Northridge, CA Quake, 1994


 * 7.0**- 32 million tons Hyogo-Ken Nanbu, Japan Quake, 1995; Largest thermonuclear weapon


 * 7.5**- 160 million tons Landers, CA Quake, 1992


 * 8.0**- 1 billion tons San Francisco, CA Quake, 1906


 * 8.5**- 5 billion tons Anchorage, AK Quake, 1964


 * 9.0**- 32 billion tons Chilean Quake, 1960


 * 10.0**- 1 trillion tons (San-Andreas type fault circling Earth)


 * 12.0**- 160 trillion tons (Fault Earth in half through center,

OR Earth's daily receipt of solar energy)

Questions:

1. According to the table what magnitude earthquake releases a similar amount of energy to small

nuclear weapon? - **A magnitude of 4.0 would release the same amount of energy as a small nuclear bomb according to the table.** 2. Approximately how many tons of energy was released during the August 23, 2011 earthquake off

centered in Virginia? - **It would have released around 90,000 tons of energy because it is between 5.5 and 6.0 on the scale.** 3. Approximately how many tons of energy was released during the March 11, 2010 earthquake off of

Japan? (8.9 on the Richter scale) - **Since 8.9 is nearly 9.0 on the scale the quake would have released around 32 billion tons of energy.** 4. According to the chart, how many times greater is a 5.0 magnitude quake than a 4.0 magnitude

quake? - **It is about 32 times larger. The two values are 1,000 tons and 32,000 tons. If you multiply 1,000 by 32 you get 32,000.** 5. According to the chart, how many times greater is an 8.0 magnitude quake than a 7.0 magnitude

quake? How many times greater is a 9.0 magnitude quake than an 8.0 ma - **It is 31.25 times larger between 7.0 and 8.0 and 32 times larger between 8.0 and 9.0. This pattern repeats towards the end of the table.** 6. Using the information from the table create a graph that gives the energy yield for any Richter

magnitude. Put Richter magnitude on the x-axis and put energy yield on the y-axis. 7. Approximately how many tons of energy was released during the 2010 earthquake in Haiti (7.1)? - **About 47 million tons according to the table.** 8. Approximately how many tons of energy was released during the 2010 earthquake in Chile (8.8)? - **According to the chart it would produce around 25 billion tons** 9. Approximately how many tons of energy was released during the 1906 earthquake in San Francisco
 * - Please See Separate Sheet of Paper for Graph**

(7.9)? - **Around 1 billion tons according to the chart.** 10. Using either your graph or the table write an exponential equation that gives the energy yield for any

Richter magnitude. **- M = 2/3log(E/Eo) M= Magnitude E= the seismic energy released Eo= the energy released by reference quake. ** 11. An earthquake has a seismic energy release of approximately 500 billion tons. About what magnitude

earthquake was this?
 * - Approximately a 9.5 magnitude earthquake.**

Graphing Quadratic Functions in Standard Form: F(x)= a(x-h)^2 +k: standard form for graphing Example: f(x)= (x-3)^2-1=8 Step 1: the parabola opens up… a=1 a>1 Step 2: determine vertex… (3,-1) Step 3: find the y-intercept… f(0)= (-3)^2-1=8 (0,8) corresponds to the y intercept Step 4: find the x-intercepts… f(x)= (x-3)^2-1=8 = (x-3)^2=1 = x=2 x=4 Step 5: plot

Graphing a Quadratic Function given in Standard form with a negative leading coefficient: Use same steps above to find points With the negative coefficient you must reflect over the x axis and vertically stretch by the factor A.

Graphing a quadratic function given in general from with a negative leading coefficient: F(x)=-3x^2+6x+2 Step 2: group… (-3x^2+6x)+2 Step 3: -3 common factor… -3(x^2-2x)+2 Step 4: add subtract 1 from both sides.. -3(x^2-2x+1-1)+2 Step 5: regroup sides: 3(x^2-2x+1)-3(-1)+2 Step 6: write the expression inside the parentheses as a perfect square and simplify… -3(x-1)^2+5 then graph.

Finding the equation of a Parabola: F(x)=x^2 and y=x^2 have the same graph Use point (2,3) to find a Step 1= f(2)= a(2-3)^2+4=3 A(-1)^2+4=3 A+4=3 A=-1 Standard form= -(x-3)^2+4 General form= -x^2+6x-5

Ryan Sullivan November 2, 2011 Precalc **__H a l l o w e e n__** Americans are expected to shell out almost $6-billion this Halloween and more than a third of that will be spent on costumes. This year is expected to be a big year for Halloween! On average, Americans will spend about $25 on their costume. An estimated 2 out of 5 Americans are expected to wear costumes this year. That's up a third from last year.

1. According to the information above, about how much money are Americans expected to spend on Halloween costumes this year? 2. According to the information above, an estimated two out five Americans are expected to wear a costume this year. If there are about 300,000,000 people living in the U.S., about how many of them will be wearing a costume this Halloween ? 3. In problem one you found the estimated amount that Americans will spend on Halloween costumes this year. In problem two you found how many Americans are estimated to wear a costume. Use this information to find how much the average Halloween costume wearing American will spend on their costume. Show all of your work/process/calculations below.
 * 1/3(6,000,000,000)= $2,000,000,000 will be spent by Americans this Halloween.**
 * 2/3(300,000,000)= 120,000,000 people will be wearing a costume this Halloween.**
 * 2,000,000,000/120,000,000= $16.70 will be spent per person on their costumes.**

4. Did your answer from problem three match the amount given at the top of the page (the information at the top of the page says that American’s will spend an average of $25 on their  costume)? 5. According to the information above two out of every five Americans will dress up for Halloween. The reading also tells us that this up one third from last year. Use this information to write the ratio of Americans wearing a Halloween costume to total Americans for the year before this year. Once you have your ratio, try writing the ratio in different ways. Which one seems the best way to write the ratio? Why? 6. If one third more Americans wear a Halloween costume next year than this year, what will be the new ratio of American dressing for Halloween to total Americans? Once you have your ratio, try writing the ratio in different ways. Which one seems the best way to write the ratio? Why? 7. If this pattern continues, each year one third more Americans wearing a Halloween costume than the year before, when would virtually all Americans be wearing a costume for Halloween? Clearly show how you found your answer. /**40,000,000**
 * It did not match.**
 * 2/5 dress up x 1/3= 2/15(300,000,000) ratio= 120,000,000: 80,000,000**
 * Or: 3:2**
 * Ratio= 300,000,000: 160,000,000**
 * Or: 15:8**
 * 40,000,000x+160,000,000= 300,000,000**
 * -160,000,000 -160,000,000**
 * 40,000,000x=140,000,000**
 * X= 3.5 years from then.**

Ryan Sullivan November 2, 2011 Chapter 1

1.2

9 different functions: - Linear; neither unless y=x -constant- f(x)=c; even - Identity- f(x)=x, odd -Square- f(x)=x^2; even - cube- f(x)=x^3; odd - square route- neither - cube route- odd - abs value- even - rational- odd

Average rate of change - to find average rate of change draw a line through two points, a secant line - find slope of line then use average rate of change formula __f(x2)-f(x1)__ dkad x2- x1

- Difference quotient

1.3

- This section will be how to graph the transformations of the graphs we learned in 1.2 - ex- x^2+3 would be a transformation of the vertex up 3 points - horizontal shifts: f(x+c) you would move the vertex c units left f(x-c) c units right

Reflection about the axes: - -x^2 would be a reflection about the x axis

Horizontal stretching: - Nonrigid transformations distort the shape of the orignal graph while rigid transformations are horizontal, vertical shifts only change the position of the graphs

1.4

Adding, subtracting, multiplying, and dividing functions - any number that is the domain of both the function is in the domain of the combined functions - the exception to this is the quotient which also eliminates values that make the denominator equal to 0 - composition of of functions when the output of one function is the input of another function - important to realize that there are two filters that allow certain values of x into the domain

1.5 - each input in the domain corresponds to exactly one ouput in the range and no two inputs map to the same ouput - Discrete points- for set of all point (a,b) verify that no y values are repeated - algebraic functions- let f(x1) = f(x2) if it can be shown that x1=x2. then the function is 1 to 1 - graphs- use the horizontal line test, if any horizontal line intersects the graph of the function in more than one point than then the function is 1-1

Determine whether a relation is a fuction - A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in the second set, called the range. - A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range. Determine whether an equation represents a fuction - The requirement for an equation to define a function is that each element in the domain corresponds exactly to one element in the range. Use function notation - We can select x-values (input) and determine unique corresponding y-values (output). Find the value of a function - To find the value you can plug in x for the input and solve the equation to find the corresponding y-value Determine the domain and range of a function - Domain is the value of x’s within the function graphed. The highest and lowest values Think of function notation as a placeholder mapping. What does this mean? - This means that by using the equation you can plot a specific point on the graph. All functions are relations, but not all relations are functions. - This is because the each element in the domain must correspond to exactly one element within the range, but with a relation a value in the domain can go with multiple values of the range.

Ryan Sullivan Precalc September 13, 2011

5 Uses of Quadratic Functions in Real Life…


 * 1) Quadratic functions can be used to find the stopping distance of a car during production research.
 * 2) A quadratic function can be used in ballistics. Forensic scientist may use these functions to calculate the path of bullets in crime scenes.[[image:http://upload.wikimedia.org/wikipedia/commons/b/bd/FN_57_ballistics.gif width="402" height="242"]]
 * 3) Using a quadratic function you can find the perfect trajectory to kick a rugby ball to complete a drop kick.[[image:http://t1.gstatic.com/images?q=tbn:ANd9GcRbr8lM-wx9ffoCuqnLcm9uqLhj4ceFvmtQcaRWcVBHy1cSV28B]]
 * 4) The military can use quadratics to make missile calculations. The parabolic path that the missile travels can be found by a quadratic equation.[[image:http://www.media4math.com/images/AlgFlipcharts27.png width="297" height="221"]]
 * 5) Economist use quadratic functions to chart progress; ups and downs in things like revenues. They can find out when things become profitable using quadratic functions.

Pictures found at: 1. www.commons.wikimedia.org 2. live-production.tv 3. media4math.com