My+Parabola+Project

My topic for my project will be the center water jet parabola of the Buckingham Fountain in Chicago.



The Buckingham Fountain of Chicago opened in 1927. The 4 seahorses represent the 4 different states that touch Lake Michigan. Those states are: Wisconsin, Indiana, Illinois, and Michigan. These seahorses are an Art-Deco style of sculpture. This fountain is an exact but bigger replica of the one at the Castle of Versailles. It symbolizes lake Michigan. The fountain uses 15,000 gallons of water per minute. The fountain runs from 8 AM until 11 PM every day from Mid-April to Mid-October. http://web.me.com/paulscott.info/maths-gallery/4/62.water-jet.html

Here is my graph of the Buckingham Fountain in Chicago:

The way I was able to graph this parabola was that I found 3 points on the graph. I did a bit of research and I found that the diameter of the largest basin of the fountain was 103 ft. Also, I found out that the center water jet shoots 150 feet into the air and still stays within the basin. Since that was the case, i divided 103 by 2 and got 51.5 ft. for the span of the water jet. My first point was (0,0) Since I knew that the parabola's vertex was at half of the span and also 150 ft high, I divided 51.5 by 2 and got 25.75. That is how I got my vertex coordinates which are (25.75,150). Then my third and final point I got by knowing that 51.5 was the X coordinate and since there isn't anything for the Y value, it is zero. so my third point was (51.5,0). This how I graphed my parabola. Equations: Here are the standard form and the vertex form of the equation of my parabola. Standard form: y=-0.2262x^2+11.65x Vertex form: y-150=-2262(x-25.75)^2 The y-intercept and vertex of my parabola: The y-intercept is (0,0). The vertex of my parabola is (25.75,150). This represents the maximum height that the water goes before it starts going back down. Factored form of my equation: y=0.2262(x-0)(x-51.5) The discriminant of my parabola: The discriminant is 11.65^2-(4*.2262*0)=135.7225. This discriminant, which is positive, tells me that my parabola has 2 solutions. Since my discriminant isn't a perfect square, it can't be factored over the integers. The x intercepts of my parabola: The 2 X intercepts of my parabola are 0 and 51.5. The 0 represents the start of the water jet. The 51.5 represents the span of my parabola and also where the parabola hits the water basin. I could find them without looking at the graph because I used the factored form of my equation to identify them. Using the Quadratic formula: A wooden board is placed horizontally 14 ft above the start of the water jet. The board is also 14 ft. tall. where will this rod be able to be placed?
 * Since the board is 14 ft, y=14 in our equation
 * Now we have our standard equation of 14=-0.2262x^2+11.65x
 * To be able to use the quadratic formula, we need y to equal 0 so we subtract 14 to both sides of the equation.
 * The equation we end up with is -0.2262+11.65x-14
 * Now we know that a=-0.2262 ; b=11.65 ; and c=-14
 * We can now use the quadratic formula to solve for x.
 * -b^2±√b^2-4ac all over 2a
 * -11.65^2+√135.7225/2•-0.2262=236.78 and -11.65^2-√135.7225/2•-0.2262=281.245
 * So the board can be placed at either about 237 ft. or at about 281 ft.