...Logarithm * Mathematical * Exponential Growth * Exponential Decay Question 1) Change the following from exponential form to logarithmic form (1 mark each): a) b) Question 2) Change the following from logarithmic form to exponential form (1 mark each): a) b) Question 3) Solve for WITHOUT using a calculator. Show all of your work. (Hint: Use the definition of a logarithm.) (2 marks each) a) b) c) d) Question 4) Apply the Change of Base Formula to rewrite the logarithms with the common logarithm. (1 mark each) a) b) Question 5) Solve for the variable. Show all of your work and all of your steps. (Hint: Use the properties of logarithms.) (4 marks each) a) b) c) d) Question 6) Solve for the variable. Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the common logarithm.) (4 marks each) a) b) c) Question 7) Solve for . Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the natural logarithm and the definition of a logarithm.) (4 marks each) a) b) c) Question 8) Ms. Mary bought a condo for $145 000. Assuming that the value of the condo will appreciate at most 5% a year, how much will the condo be worth in 5 years? Section 2: Conic Sections Standard forms to Know: * Parabola * Circle * Ellipse * And what does a hyperbola look like? (No formula necessary) Question 1) Write an equation for the circle that satisfies...
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...Maths in nature "The laws of nature are but the mathematical thoughts of God" - Euclid Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted in nature. SYMMETRY Symmetry is everywhere you look in nature . Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. There are two kinds of symmetry. One is bilateral symmetry in which an object has two sides that are mirror images of each other. The human body would be an excellent example of a living being that has bilateral symmetry. The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn. The most obvious geometric example would be a circle. Shapes Sphere: A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator. The wee electron has gotten its most thorough physical examination yet, and scientists report that it is almost, almost a perfect...
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...New York State Common Core Mathematics Curriculum GEOMETRY • MODULE 1 Table of Contents1 Congruence, Proof, and Constructions Module Overview .................................................................................................................................................. 3 Topic A: Basic Constructions (G-CO.1, G-CO.12, G-CO.13).................................................................................... 7 Lesson 1: Construct an Equilateral Triangle ............................................................................................. 8 Lesson 2: Construct an Equilateral Triangle II ........................................................................................ 16 Lesson 3: Copy and Bisect an Angle........................................................................................................ 21 Lesson 4: Construct a Perpendicular Bisector ........................................................................................ 30 Lesson 5: Points of Concurrencies .......................................................................................................... 37 Topic B: Unknown Angles (G-CO.9) ..................................................................................................................... 43 Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point ....................................................... 44 Lesson 7: Solve for Unknown Angles—Transversals .................................
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...sixty proofs of the Butterfly Theorem, including the synthetical proof, area proof, trigonometric proof, analytic proof and so on. And based on the extension and evolution of the Butterfly Theorem, people can get various interesting and beautiful results. The definition of the Butterfly Theorem is here below: “Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.” (Bogomolny) This is the most accurate definition currently. However, Butterfly Theorem has experienced some changes and developments. The first statement of the Butterfly Theorem appeared in the early 17th century. In 1803, a Scottish mathematician, William Wallace, posed the problem of the Butterfly Theorem in the magazine The Gentlemen’s Mathematical Companion. Here is the original problem below: “If from any two points B, E, in the circumference of a circle given in magnitude and position two right lines BCA, EDA, be drawn cutting the circle in C and D, and meeting in A; and from the point of intersection A to the centre of the circle AO be drawn, and the points E, C; B, D joined, and produced to meet an indefinite perpendicular erected at A on AO; then will FA be always equal AF. Required the demonstration?”(Bogomolny) (Figures of W Wallace’s question) Soon afterwards, there were three solutions published in 1804. And in 1805, William Herschel, a British...
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...The children got drop off at school, and one-by-one, start to get on the line to walk into their classroom. At the classroom, each child took his coat off and put it away with their bag pack on their cubby. When finish each child look on a table for his or her file and it wrote his/her name. The children proceed to wash their hand and set on the circle time area for circle time. At circle time the teacher asks each child about their weekend, and one-by-one describe all the activities they did. They sing and play respect to the flag, went over the calendar, the weather, and assigned the duties of each child for the week. After circle time the children wash their hands and start to get ready for lunch, the children who have to help this week on lunch and snack went to the kitchen with the teacher and help brought the lunch, and set up the table. The children and teacher set at the tables and eat lunch. The children have the opportunity to decide what and how much to eat. When they finish the children wash their hands and the teacher told them it was going to be reading time. Each child went to the reading area of the classroom and choose a book to read, each child choose were to set to read. The class went to the gym to play because it was a raining day and they can go to play outside. The teacher provides the children with tricycles, and balls, so they can play, some children play basketball, others soccer, some drove the tricycles, and others just ran all over the gym. When the...
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...the result is a circumference. lines draws 9 inner circles, with radius equal to 1 * i, i ranging from 0.1 to 0.9 in sequence. lines apply the shrinking factor i because it relates to i by means of the for loop. The same structure is used to assign numbers to circles, through the function text. Target face and probabilities a) What is the probability of hitting any possible point within the inscribed area of radius ½? To draw the inscribed circle, apply a shrinking factor to both sin(a) and cos(a) in polygon function: > polygon(.5*sin(a),.5*cos(a),col="red") Dart game – CompTools2012 user guide In this user guide we are going to solve some problems related to the dart game. Follow us step by step and enjoy the game! We want to build a target face for throwing darts, like the one in the figure. We assume that it is a circle of radius 1 with center in the origin point. For the construction of the target, the reference is paragraph 6.2 of Using R for scientific computing by Karline Soetaert. We present here the necessary script: > a plot(cos(a),sin(a),type="l",lwd=2,xlab="" ,ylab="",axes=T, asp=1) Ten Inner circles at equal distance: > for (i in seq( 0.1,0.9,by=0.1)) lines(i*sin(a), i*cos(a)) >polygon(sin(a)*0.1,cos(a)*0.1,col="red") > for (i in 1:10) text(x = 0, y = i/100.025, labels = 11-i, font = 2) >for (i in 1:9) text(x = 0, y = -1 + (i/10-0.025), labels = i, font = 2) The polygon command will cover the inner circles: launch again the previous for loops to obtain the...
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...Worksheet for Week 1: Circles and lines This worksheet is a review of circles and lines, and will give you some practice with algebra and with graphing. Also, this worksheet introduces the idea of “tangent lines” to circles. Later on in Math 124, you’ll learn how to ﬁnd tangent lines to many other types of curves. 1. Two circles, called C1 and C2 , are graphed below. The center of C1 is at the origin, and the center of C2 is the point in the ﬁrst quadrant where the line y = x intersects C1 . Suppose C1 has radius 2. C2 touches the x and y axes each in one point. What are the equations of the two circles? y y=x C2 x C1 Worksheet Math 124 Week 1 2. Let C be the circle of radius 5 centered at the origin. The tangent line to C at a point Q is the line through Q that’s perpendicular to the radial line connecting Q to the center. (See picture.) Use this information to ﬁnd the equations of the tangent lines at P and Q below. y Q P x Note: Later in Math 124, you’ll learn how to ﬁnd tangent lines to curves that are not circles! Page 2 Worksheet Math 124 Week 1 3. Sketch the circle of radius 2 centered at (3, −3) and the line L with equation y = 2x + 2. Find the coordinates of all the points on the circle where the tangent line is perpendicular to L. y x Page 3 Worksheet Math 124 Week 1 4. Draw the circle with equation x2 +y 2 = 25 and the points P = (−3, −4) and Q = (−8, 0). Explain why P is on the circle. Is the line through P...
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...the third shape: ") Step 4) next we need to create and print out a list of the three shapes the user inputed: so we type: list = [shape1, shape2, shape3] print(list) So, the first half (list of three shapes) should look like this: import math print("Let's create a list of three shapes...") shape1 = input("Enter first shape: ") shape2 = input("Enter the second shape: ") shape3 = input("Enter the third shape: ") list = [shape1, shape2, shape3] print(list) Step 5) Lets move on to the second half: we need to find the are and circumfrence of a circle. First we need to know the formula (or algorithm) to find the area/ circumfrence of a circle. Step 6) We need to tell the user that we are going to find the area and circumfrence of a circle... so we say: print("\nNow, let's find the circumference and area of a circle...") Step 7) Next we need to get the user to input the radius of the circle that they want to find the circumfrence and area for: we need the radius to be entered as a specific type of data called a float (a number with decimals) so we type: rad = float(input("Enter radius: ")) Step 8) Now we need to compute the circumfrence: we now...
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...The symbols in each movie 1) The photos in BLOOD SIMPLE are the start of all the misunderstandings. In the past Marty didn’t know anything about the fair of his wife. But after the first few pictures taken by Visser, his anger came out and he started his revenge and got involved in this nightmare. The fake picture of Abby and Ray dead in the bed also misleads Marty. When Marty was about to give Visser the money, Visser shot Marty. And Visser throw Abby’s gun in the room to create a misleading scene. These things start with the fake picture. And as Visser says, “life is very uncertain and something can always go wrong.” The appearance of the fake pictures is the moment that life goes wrong for Marty. Marty’s life was totally changed because of this fake picture. He wanted to kill Ray and his unfaithful wife, but instead he is the first one who was shot and got buried alive. Because of the photos, a series of misunderstandings begin. Ray thought Abby killed Marty and Abby thought Visser was Marty and killed him. All the twists start from these photos. Everybody was confused after this. Nobody in the movie truly knows what was happening to everybody. Even Visser, who thinks he knows everything, was also confused at the end. The photos are fake, and this tells us that everything can happen because of money. The photos can be faked, and also the relationship and everything valuable in the world can be faked and changed because of money. The photos triggered the twists and misleading...
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...think on what to do with our second half of lives because we cannot put hope that we will still be working in the same organization in twenty years from now. As the saying goes, “Failure to prepare is preparing to fail.” We were taught in class the graph of life; as we are getting older, the options that are available to us are getting smaller, but the responsibilities are getting bigger. So at this young age, we need to grab any chances for the benefit of our future. As the professor said in class, “Don’t get old, alone and broke.” If we did, we might have nothing at all; even family. We do not want to be a burden to ourselves and people around us. Life is about drawing circles. If we have a great life, we will draw a perfect round circle. If we slipped somewhere, the circles drawn might not look like a circle. There are three ways to spend the second half of life. The first is to start a second and different career which is moving from one kind of organization to another. They may stay in the same kind of work, but some move into a different line of work. These people are actually happy with their first job but they need a community as the kids are grown up. They need income too but most...
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...ITT TECHNICAL INSTITUTE GS1145 Graded Assignments Unit 1 Assignment 1: Change Wheel Course Objectives and Learning Outcomes Describe personal changes in relation to global/historical changes. Communicate information using Microsoft Office productivity tools and email. Assignment Requirements Review Chapter 1, pp. 4-19 and then complete the Change Wheel Worksheet (found on the next page in this graded assignment.) Required Resources Completed Preparation for Success Checklist Textbook Submission Requirements Submit completed worksheet to instructor by the beginning of Unit 2. The Change Wheel Personal and Global Perspectives Before completing this assignment, please review Chapter 1, pp. 4-19. In Unit 1, we looked at change from several vantage points, including need, difficulty, and strategies for success. To complete this assignment, think carefully about change you have experienced in your personal life, and change in the world around you during your life. Task 1: Personal Change Identification Directions: Brainstorm a list of significant changes you have personally experienced throughout your life. Review your list, and identify six major changes that have affected you in very important ways. List them here: 1. Paying Car note 2. Changing Friends 3. Keeping Postive Attitue 4. Stop Complaining 5. Becoming Independent 6. Looking and Talking Professional 7. __________________________________________________________ ...
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...to be equal the from the left side add the technic bush (gray). Then push the technic axle 8 inside. Step 1 : First we bring the electric mindstorms NXT motors and add to it the small technic pin . after you add it add the technic axle 8 in the middle to be equal the from the left side add the technic bush (gray). Then push the technic axle 8 inside. Step 2 : Second we bring the technic beam 3 and we put it inside the technic axle then add we add the technic bush (yellow) to the technic beam 3 after that we put the technic pin (small) in the 3 circles put the last two then down the technic bush (gray)add the technic beam 13 and put the technic pin (long ) the last 3 in the right and the first one on the left. Step 2 : Second we bring the technic beam 3 and we put it inside the technic axle then add we add the technic bush (yellow) to the technic beam 3 after that we put the technic pin (small) in the 3 circles put the last two then down the technic bush (gray)add the technic beam 13 and put the...
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...Surname Centre No. Candidate No. Paper Reference(s) Initial(s) Paper Reference Signature 1 3 8 0 1380/4H 4 H Examiner’s use only Edexcel GCSE Mathematics (Linear) – 1380 Paper 4 (Calculator) Team Leader’s use only Circle Theorems Past Paper Questions Arranged by Topic Materials required for examination Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used. Items included with question papers Nil Instructions to Candidates In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper. Answer ALL the questions. Write your answers in the spaces provided in this question paper. You must NOT write on the formulae page. Anything you write on the formulae page will gain NO credit. If you need more space to complete your answer to any question, use additional answer sheets. Information for Candidates The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 26 questions in this question paper. The total mark for this paper is 100. There are 24 pages in this question paper. Any blank pages are indicated. Calculators may be used. If your calculator does not have a π button, take the value of π to be 3.142 unless the question instructs otherwise. Advice to Candidates Show all stages in any calculations. Work steadily through the paper...
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... Bus The “Bus Topology” is a network setup in which each computer and network device are connected to a single cable. This type of setup is not good for large networks for many reasons. Some reasons are difficulty in troubleshooting individual devices and the entire network, the network can fail if the cable is damaged, and the more devices on the network slows down the entire network. In contrary to larger networks, this set up is perfect for smaller networks, because it requires less cable length and setting up the actual network is easier as well. Ring This network is mostly used in schools and offices, where the networks are smaller. In this setup the devices are connected to each other going in a circle shape, so that each packet must go around the circle until it reaches its destination. Data flows only in one direction at high speeds between the workstations, therefore no requirement for a server needed. In addition, the cost is a bit more expensive to connect the workstation and when a workstation shuts down the entire network will be...
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...which easies some of her distress as she is aging knowing one day that the memories of her youth will soon fade. The long strokes of lines and curves in the mirror by her face and around her head display a covering a sense of innocence; before she has known a man intimately. The lines in the lower area of her body in the mirror shows it is still in development as the lines are arched up and not down in a drooping manner as with age. Her body has not set firmly in position so it appears altered in the mirror just like a teenager starting puberty; one breast grows larger than the other. In her conversation with herself she knows her innocence has vanished, and a development is changing her life once again; the baby growing within her. The circles remind me of life. There is a beginning and an ending....
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