Κυριακή, 16 Ιανουαρίου 2011

Golf ball

A golf ball is dropped and a regular strobe light illustrates its motion as follows...
 We observe that it is a parabola.

Problem:What is the equation of the parabola that the golf ball is tracing out?


ΓΙΑ ΤΟΥΣ ΜΑΘΗΤΕΣ: χρησιμοποιήστε το Excel για να σχεδιάσετε το γράφημα.
Use Excel for making the graph and finding out the equation!


We can model the motion from the data points obtained by observing the position (height) of the ball at various times.
This is the table of values time/height:
math expression
Source: http://www.intmath.com/Plane-analytic-geometry/4_Parabola.phphttp://wn.com/The_Ball_and_the_Parabola

Also the tennis ball does the same motion. Watch the video:
____________

Using Excel to Model Curves



You can use Microsoft Excel to module a parabola. After you plot the points, right-click on one of the points and choose "Add Trendline (Γραμμή Τάσης)".
Choose Polynomial, degree 2. In "Options (Μορφοποίηση γραμμής τάσης --> Επιλογές)" you can get Excel to display the equation of the parabola on the chart.

Is the Gateway Arch a Parabola?

The Gateway Arch, or Gateway to the West, is part of the Jefferson National Expansion Memorial in St. Louis, Missouri.

Built as a monument to the westward expansion of the United States, the arch has become the iconic image of St. Louis. It typifies "the pioneer spirit of the men and women who won the West, and those of a latter day to strive on other frontiers". At 630 feet (192 m), it is the tallest man-made monument in the United States, and Missouri's tallest accessible building. Located on the west bank of the Mississippi River.

Why do they use the parabola shape for bridges ?
The arch helps transfer force to the outside of the bridge so that the abutments can support it.

Comparing our parabola to the photo, we can see (especially near the waterline at the bottom) the shape of the arch is not really a parabola at all. The designer of the Gateway Arch, Eero Saarinen, knew that a parabola was not the best shape for such an arch. An arch shape that is often used is the catenary. The Gateway Arch is also the largest architectural structure designed as a weighted or flattened catenary arch.

So, the Gateway Arch is not a parabola. It is a catenary arch!!!



Gateshead Millennium Bridge in Newcastle

Blinking eye bridge



The bridge takes its place at the end of a line of distinguished bridges across the River Tyne, including the Tyne Bridge and Robert Stephenson's High Level Bridge.
Linking Gateshead with Newcastle via Gateshead Quays (described as one of the best places in Europe by Tony Blair) and Newcastle's Quayside, the bridge not only serves a functional purpose as the River Tyne's only foot and cycle bridge, but its grace and engineering attract people from all over the world. 
The 130m long deck is parabolic in elevation and of steel box section that tapers in plan towards the centre of the deck. It carries a pedestrian footway that varies from 3m to 5m in width as well as a 2.5m cantilevered cycleway. 

The main arch is also parabolic in shape and tapers both in plan and elevation.
Six 450mm diameter Hydraulic rams (three on each side, each powered by a 55kW electric motor) rotate the bridge back on large bearings to allow small ships and boats (up to 25m tall) to pass underneath. The bridge takes as little as 4.5 minutes to rotate through the full 40° from closed to open, depending on wind speed. Its appearance during this manoeuvre has led to it being nicknamed the "Blinking Eye Bridge".

Arches

Κυριακή, 9 Ιανουαρίου 2011

Find the equation of the quadratic using Geogebra

Use the application of Geogebra (parabola_water.html) and find the equation of the .... dancing waters of the fountains of Montjuic in Barcelona!!

(Geogebra will load here. Please be patient as it takes a short time for the Java to load!)
Move the sliders to position the parabola over one of the red fountains.Let's go...

Find the equation of the quadratic function


Using the excel file, solve the exercises below:

Find he equation of the quadratic functions:

Using Geogebra, make sure that your solutions are right!
Capture your results and save it as image (jpg file)!!!

Πέμπτη, 6 Ιανουαρίου 2011

World War II Memorial

The World War II Memorial in Washington D.C.honors the 16 million who served
in the armed forces of the United States during the war and the more than 400,000
who died. It is particularly beautiful at night as the following photos show.

The fountains are also an excellent example to be used in mathematics
as you will see the Geogebra application below: 


Click on the fountain to analyze the parabola
of the dancing water
Please be patient as it takes a short time for the Java to load! 




Gaudi's parabolic arches

Parabolic arches, Barcelona, Spain
Beautiful Barcelona and the genius of Gaudi

These are catenary arches!


Under the roof of Gaudí's Casa Milá, Barcelona, Spain

The catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force
(its own weight).





Don't confuse! These are not parabolic arches!


Arches in Casa Mila, by Gaudi too!

The beloved.... parabola of Gaudi

The peaceful area inside the gardens of the Parc Güell

in Barcelona

and the Gaudi's beloved  parabola .

Τρίτη, 4 Ιανουαρίου 2011

Parabola Monument in Barcelona

The Fossar de les Moreres (Moreres Cemetery) is a memorial plaza in Barcelona (Catalonia, Spain), adjacent to the basilica of Santa Maria del Mar. The plaza was built over a cemetery where defenders of the city were buried following the Siege of Barcelona at the end of the War of the Spanish Succession in 1714.

The plaza retains its everyday use as a public space, but also prominently features a memorial to the fallen Catalans of the war, with a torch of eternal flame and a heroic poem by Frederic Soler, "El Fossar de les Moreres". On 11th September, in this place, the Catalan people celebrate their National Day. This historical date refers back to September 11 in 1714, when Barcelona, following a 13-month siege, surrendered to the Castilian and French soldiers.

The Monument

Look at the monument!
Keep in mind that it has the shape of a parabola!
Which of the following functions corresponds to this parabola? 
  • Y = x2 + 2x – 4
  • Y= - x2 + 7x
  • Y= -x2 + 4
 



Look at:

Δευτέρα, 3 Ιανουαρίου 2011

Positive - negative quadratic - Vertex

positive quadratic y = x2
negative quadratic y = - x2

 

If the coefficient of x2 is positive the curve looks like the mouth of a smiling child.

The vertex (0,0) is down.
If the coefficient of x2 is negative the curve looks like the mouth of a joyless child.

The vertex (0,0) is up.



Source: The purplemath


The meaning of a in a quadratic


The general form of a quadratic is:
y = ax2 + bx + c

For graphing, the leading coefficient "a" indicates how "fat" or how "skinny" the parabola will be.

For | a | > 1 (such as a = 3 or a = –4), the parabola will be "skinny", because it grows more quickly (three times as fast or four times as fast, respectively (in the case of our sample values of a). 

For | a | < 1 (such as a = 1/3 or a = –1/4 ), the parabola will be "fat", because it grows more slowly (one-third as fast or one-fourth as fast, respectively (in the case of our sample values of a). 

Also, if a is negative, then the parabola is upside-down.






You can see these trends when you look at how the curve y = ax2 moves as "a" changes.
Click on the adjacent image (graph) -->










Source: The purplemath

2 straight lines

Let's plot 2 straight lines:
  • y=3x+b
  • y=-3x+b

Look at the Geogebra application below.
Write your remarks?
....................................................................................

The linear equation

ψ=aχ+b


Any time you have a "ψ equals a number" equation (i.e. y=5, with no x in it), the graph will always be a horizontal line.
...........................................................................................


Any time you have an equation as ψ=aχ+b with a<0, then its graph is a straight line  and the corresponding function f(x) = aχ+b is decreasing.

Changing the value of a, the angle between the x-axis and the straight line is changing.
Have a look at Excel file: y=-aχ+b.xls (Click here)
...........................................................................................



Any time you have an equation as ψ=aχ+b with a>0 , then its graph is a straight line and the corresponding function f(x) = aχ+b is increasing.
Changing the value of b, the straight line is moving from left to right and vice versa across the axis x, changing the y-intercept.

...........................................................................................
WORK
1. Have a look at the Excel file revised y=aχ+b.xls (Click here) and move the scrollbar in order to change the value of a and b.
What do you remark?
...........................................................................................

2. Go to the sheet Sales of the same excel file.
What's about the line in the graph?
...........................................................................................


Which is the straight line that shows that you have bought cheaper  and why?
...........................................................................................

Coordinate System - Glossary

Important things to remember:

• All points located in Quadrant I have positive x- and y-coordinates.
• All points located in Quadrant II have negative x-coordinates and positive y-coordinates.
• All points located in Quadrant III have negative x- and y-coordinates.
• All points located in Quadrant IV have positive x- coordinates and negative y-coordinates.
• All points located on the x-axis have the y-coordinate equal to zero.
• All points located on the y-axis have the x-coordinate equal to zero.

In this video, you will learn how to plot various points in the coordinate plane (system).

Source: Video 

Work:
Please remember to fill in the cells of the shared google spreadsheet: Coordinate system Glossary.
You need to login with your google (gmail) account!

Σάββατο, 1 Ιανουαρίου 2011

Happy New Year 2011

May all the dreams in your eyes,
all the desires in your heart and
all the hopes in your life blend together,
to give you the most spectacular New Year ever
Happy New Year 2011!















 May all the dreams in your eyes,
All the desires in your heart and
All the hopes in your life blend together,
To give you the most spectacular New Year ever
Happy New Year 2011!