...Bridge – Directed Investigation – Quadratic functions Introduction: Aim: To find the multiple unknowns in the Sydney Harbour bridge The Sydney Harbour Bridge will be used to investigate a diverse number of points in the structure such as the height and length. Quadratics will be used to solve the height of the bridge at different points on the x axis. A quadratic is an equation constructed from information collected from a graph; this equation can also be used to produce a graph. Quadratics can be used to solve the problem dealt in this investigation as the Sydney Harbour Bridge is identified as a parabola shape. Only basic information is given about the bridge and the answers can be established by solving the additional questions. The quadratic equation will be tested from the graph to prove the accuracy of the quadratic identified. Quadratics are mainly seen in the form ax^2+bx+c=0, where x is a variable of any number. The bridge is mounted on two base pylons on the opposite ends. The highest point of the bridge is 182.5cm above sea level and the longest vertical cable is 135m from the origin. This position is the vertex, which is the highest or lowest point inside a quadratic, and in this case it is highest in this scenario. The axis of symmetry is an imaginary vertical line that cuts through and splits both sides evenly on the graph. The main supporting arc has 2 points that sits on the road situated 50m above sea level. Mathematical Investigations: The bridge is supported...
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...The Ellipse Definition of Ellipse Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2a. General Equation of the Ellipse From the general equation of all conic sections, A and C are not equal but of the same sign. Thus,the general equation of the ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 or Standard Equations of Ellipse From the figure above, and From the definition above, Square both sides Square again both sides From triangle OV3F2 (see figure above) Thus, Divide both sides by a2b2 The above equation is the standard equation of the ellipse with center at the origin and major axis on the x-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above. Ellipse with center at the origin Ellipse with center at the origin and major axis on the x-axis. Ellipse with center at the origin and major axis on the y-axis. Ellipse with center at (h, k) Ellipse with center at (h, k) and major axis parallel to the x-axis. Ellipse with center at (h, k) and major axis parallel to the y-axis. The Hyperbola Submitted by Romel Verterra on February 21, 2011 - 1:33pm Definition Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci...
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...Application Activity: Angry Birds – Andrew Fox Consider the following scenario: Red Bird, Yellow Bird, Blue Bird and Black Bird are angry with the pigs who stole the birds’ eggs. The birds want their eggs back and will stop at nothing to get them back. The flight path of the birds can be modeled with a parabola where “x” is the distance and “y” is the height. Use the data below to help answer the following questions: * Red Bird starts his flight from point (10, 0). His flight path reaches a maximum height of 18 yards and lands at point (38, 0). * Yellow Bird’s flight path can be modeled by the quadratic equation y=-x2+14x-24 * Blue Bird’s flight is modeled by the following graph: * The table below contains partial data points of Black Bird’s trajectory: x | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | y | 0 | 7.5 | 14 | 19.5 | 24 | 27.5 | 30 | 31.5 | 32 | 31.5 | | In developing responses to the problems, be sure to show all work: 1. What is the maximum height of each bird’s flight: (6 points) 2. What is the axis of symmetry for each bird’s flight: (6 points) 3. What was the total distance of each bird’s flight: (7 points) 4. Which bird flew the highest? (2 points) 5. Which bird traveled the longest? (2 points) 6. Which bird hit the following pigs: a. King Pig located at point (21, 19.5) (1 point) b. Moustache Pig located at point (9, 21) (1 point) 1. Red max height (24,18) Yellow max height (7,25) ...
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...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 each set of conditions. (2 marks each) a) centre (12, -4), radius 81 units _________________________________________ b) centre (0, 0), radius 3/5 units...
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...Term Paper Mathematics NAME: BIPIN SHARMA ROLL NO: B59 SECTION: C1903 Conics Conic sections are the curves which result from the intersection of a plane with a cone. These curves were studied and revered by the ancient Greeks, and were written about extensively by both Euclid and Appolonius. They remain important today, partly for their many and diverse applications. Although to most people the word “cone” conjures up an image of a solid figure with a round base and a pointed top, to a mathematician a cone is a surface, one which is obtained in a very precise way. Imagine a vertical line, and a second line intersecting it at some angle f (phi). We will call the vertical line the axis, and the second line the generator. The angle f between them is called the vertex angle. Now imagine grasping the axis between thumb and forefinger on either side of its point of intersection with the generator, and twirling it. The generator will sweep out a surface, as shown in the diagram. It is this surface which we call a cone. Notice that a cone has an upper half and a lower half (called the nappes), and that these are joined at a single point, called the vertex. Notice also that the nappes extend indefinitely far both upwards and downwards. A cone is thus completely determined by its vertex angle. Now, in intersecting a flat plane with a cone, we have three choices, depending on the angle the plane makes to the vertical axis of the cone. First, we may...
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...Definition of Ellipse Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2a. General Equation of the Ellipse From the general equation of all conic sections, [pic] and [pic] are not equal but of the same sign. Thus, the general equation of the ellipse is [pic] or [pic] Standard Equations of Ellipse Elements of the ellipse are shown in the figure above. 1. Center (h, k). At the origin, (h, k) is (0, 0). 2. Semi-major axis = a and semi-minor axis = b. 3. Location of foci c, with respect to the center of ellipse. [pic]. 4. Length latus rectum, LR 5. Consider the right triangle F1QF2: Based on the definition of ellipse: [pic] [pic] [pic] By Pythagorean Theorem [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] You can also find the same formula for the length of latus rectum of ellipse by using the definition of eccentricity. 6. Eccentricity, e DEFINITION: Eccentricity of Conic Eccentricity is a measure of...
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...equation: p = -25x2 + 300x quadratic equation Therefore you can find the max profit by finding the value of x of the axis of symmetry and find the vertex with that: Axis of symmetry formula: x = -b/ (2a) In equation; p = -25x2 + 300x a = -25 b = 300 x = -300/ (2)-25 x = -300/ -50 x = 6 clerks will maximize the profit To find the max profit, substitute 6 for x in the original equation: P =-252+300x=0 P = -25(62) + 300(6) Substitute 6 for x. P = -25(36) + 1800 factor both left and right side. P = -900 + 1800 add to solve equation. P = $900 is the actual profit A profit/clerk graph will look like this: The basic shape of the graph in this equation of a parabola that opens downwards (coefficient of x2 is negative) so the maximum value of P will be found at the parabola's vertex. The parabola will cross the x axis at 0 and 6. To maximize profits, the manager should employ 6 clerks. The maximum profit can be found by substituting 6 for x in the original equation for P. The graph represents the maximum number of clerks needed to gain the maximum daily...
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...the cone horizontally, you are left with a circle. If you take a slice roughly at a forty five degree, you will be dealing with an ellipse. If you take a slice that is parallel from one edge of the cone to the other cone, you are dealing with a parabola. If you take a slice from directly off centered but straight down from top to bottom, you give yourself a hyperbola. These are a few terms with definitions you will see while working with conic sections. In a circle, ellipse, and a hyperbola you have a Center. Which is usually at the point of (h,k.) The focus or “Foci” is the point which distances are measured in forming the conic. The directrix is the distance that is measured in forming the conic. The major access is the line that is perpendicular to the directrix that passes through the foci. Half of the major axis between the center and the vertex is called the semi major access. There is a general equation that covers all the conic sections and goes as follows: Ax2+Bxy+Cy2 + Dx+Ey+F=0. From this equation you can create equations for circles, ellipses, parabolas and hyperbolas. There is a test to find out which conic section you are dealing with by just looking at the equations. If both variables not squared then it’s a parabola, if it is you can move on and look to see if the squared...
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...* Link Mechanism A link mechanism can be defined as a system of connecting parts/rods that move or work together when a particular action is pre-determined. * Parabola The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and equal to one. This constant is also known as the eccentricity. * Ellipse The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and less than one. This constant is also known as the eccentricity. * Hyperbola: The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and greater than one. This constant is also known as the eccentricity. * Archimedean Spiral The path traced out by a point along a rod, as the rod pivots about a fixed end. The linear movement of the point along the rod is constant with the angular movement of the rod. * Involute The path traced out by a point when an end of a plane figure is wrapped or unwrapped when held firm. * Epicycloid Tracing the path of a point as a circular disc rolls on the outside...
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...e^-0.12t = 1.85454/5 = 0.370908 -0.12t = ln 0.370908 t = ln 0.370908/-0.12 = 8.265 ≈ 8 weeks 2. According to the information (bacteria in culture after 3 hours is 100 and 5 hours is 400), the bacteria appears doubling every hour. If this is the case a) the growth rate is 200% every hour . b) Following this same trend to find the initial growth rate we would divide 100 by 2, 3 times. Thus the initial number of bacteria would be 12.5 c) Since the number of bacteria of 5 hours is 400, to find the number of bacteria at 6 hours is going to be double the amount at 5 hours, 800. 3. To solve we need to find the equation of the parabola in vertex form, since we know a point of the vertex, also that is vertically symmetrical from this point. Recall that the equation we’ll use is the quadratic y = a(x-h)² + k. Since we are looking for how wide the parabola is at 10 meters(8 meters) and its vertex’s x coordinate is at 0 and its vertically symmetrical the a point that is passed through at that height(10 meters) is 4 (4,10).Substitute this information into the equation (10=a(4-0)^2) + 12 simplify -2=16a 12/16=a a = -1/8 Now substitute the equation with a=-1/8, leave x and y as these are this is the point we are looking for and the variables we will solve for this equation is y=-1/8x^2 +12 Now find the x-intercepts at this height (0 or ground level) ((-1/8x)^2 ) +12 =y set y = to 0 ((-1/8x)^2 )+12 =0 now simplify (-1/8x)^2 = -12 x^2 = 96 x = ±√96 x=±4√6 Now subtract...
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...Parábola, según su sentido griego, el termino parábole, sugiere una comparación, es decir, para significa al lado y bole echar. Este sentido comparativo aparece en los tres evangelios sinópticos sin embargo Juan utiliza otro termino, paroimia, es esta palabra también para significa al lado, pero oimai se traduce por suponer, figurar, pensar, y tradicionalmente se ha traducido en Reina Valera 1960 por alegoría y en Reina Valera Actualizada por figura. Se puede decir que según su sentido griego una parábola es lenguaje figurado que provoca una comparación con el fin de aclarar o iluminar una cosa o idea. A la hora de leer las parábolas se deben tener en cuenta tres cosas. La Historia fijándonos en el contexto histórico y social del momento. Procurando ver la parábola dentro del marco cultural del pueblo de Palestina en los tiempos de Jesús, sobre todo los modos y costumbres. Literaria viendo la parábola como una creación literaria que respeta las normas de composición literaria en cuanto a narrativa, alegoría, retórica, etc. sin caer en una lectura de las parábolas como tratados doctrinales de teología. Finalmente Hermenéutica intenta interpretar la parábola de forma correcta y aplicarla a nuestra realidad hoy. Por esto se puede considerar que las parábolas nos hablan de la intervención de Dios en la historia. Nos retan a dar una respuesta en arrepentimiento y fe, nos invitan a buscar el reino de Dios, por eso dicen que son evangelisticas. Las parábolas de Jesús son aquellas...
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...measure of the consistency of cohesive soil and the packing of granular soil. The relationship between Mackintosh Probe and Safe Pressure is as follows: - P = (2860 + 550 (R - 40)1/2) x 0.04788 kN/m2 for blows > 40 P = Refer Chart for blows < 40 Where, P = safe pressure (kN/m2) R = Mackintosh Probe Penetration resistance in blows/0.3m For more information or site appointment, please call us or drop us a message here. The site investigation is the one thing that must be done before starting the construction of the building. This is because the soil condition at the site need to be identifies to determine the suitable foundation use for the building. As we know, soil play a main role to support the load that come from the building and the building need a suitable foundation to transfer the load to the ground. Therefore, the investigation of soil need to be done to identify the type of soil to ensure the soil can carry the load. In investigating the soil condition, the probe mackintosh commonly...
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...Johnny Brown 11/08/2012 Final Term Paper Arson Investigator The reason I decided to choose this topic, and not something about a particular part of the fire service, is because I was watching a TV show CSI where they solve crimes and such. Well this topic of fire fighter arson was on the show where a fire fighter set a fire on purpose. So I decided that it looked interesting considering it is what I want to do. Fire fighter arson is basically when a fire fighter sets a fire and I will break it down on why some fire fighters do it, affects of fire fighter arson, basic profile of the fire fighter arsonist, fire service responsibilities in preventing fire fighter arson, and actions to take when a problem is suspected. There were a number of people who had been arrested for this crime and were willing to share their motivations. The biggest reason was people were looking for recognition and liked to play the role of a hero. These people would get a thrill to be the reporting party or to be first on the scene. Another reason is a person might have a psychiatric problem that they have towards the fascination with fire. While others have a low self-esteem or self-confidence, and they want to feel like they belong to a group and are accepted by others. There are many effects of fire fighter arson, which all are very clear cut effects, they include the possibility of injury, loss of life, property and financial loss due to fire, and the increase of insurance premiums that we pay...
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... IN-SITU TESTING In-situ testing techniques including Standard Penetration Testing, Permeability Testing, Borehole Vane Testing, Pressure meter Testing and Packer Testing can all be carried out in the boreholes in order to provide information for geotechnical design. Disturbed and undisturbed samples are retrieved from the boreholes for inspection and logging by engineers and subsequent testing in our laboratories. TRIAL PITTING Trial pitting can be carried out by a variety of methods from hand dug pits to machine excavated trenches. Trial pitting is generally carried out to a maximum depth of 4.5m with standard excavation plant and, depending on soil conditions, is generally suitable for most low rise developments. All trial pit investigations are supervised by experienced engineers with a thorough understanding of geology and soil mechanics. DYNAMICS PROBE TEST Cone Penetration Test To carried out the cone penetration test is pushed into a soil deposit while various measured parameters are recorded. The test is similar to the Dutch Deep sounding (or piezocone test) with the addition of a cone penetration element in the probe to measure water pressure. The test is also known as pore pressure sounding or CPTU. Mackintosh/JKR Probe Test This is a dynamic penetrometer test used to check the consistency of the subsoil. Mackintosh Probe which has 30° cone penetrometer while JKR Probes has 60° cone penetrometer. This is a light dynamic test and the cone is driven directly...
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...The Evolution of Fire Investigation and Its Impact on Arson Cases Journal Article by Kevin Weitzel Arson investigation is an ever changing field of science and crime investigation. Arson investigators need to be highly educated in their field and unfortunately most are not properly educated resulting in many high profile cases being unfinished or citizens being wrongly convicted. This paper will cover an article by John J. Lentini discussing these topics. Fire investigation is a complex field of work that unfortunately is not drawing in college graduates because of its low pay scale. This in turn makes being a fire investigator just a matter of learning from the older more experienced fire investigators who they themselves were never properly educated. To quote Lentini (p. 1 2012) “The reality is that the fire investigation profession has within its ranks a large number of individuals who don’t know what they’re doing”. This is quite a disturbing thing to read, professional testimonies we rely on to send people away for 10, 20, or even 50 years in prison could be complete hacks, taught by hacks, and really have no idea what they’re doing but nobody else will question them either, which makes all of this a lot worse. In 1994, a group of scientists including John J. Lentini developed a position paper stating that an accelerate detecting canine (ADC) alert was in fact not suitable for a jury to hear but were in fact much more suitable as a means to prove probable cause to...
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