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PARABOLA EXPLORATION

Table of Content
Creating Functions from Parabolas
(Page 1 – 3)

Creating Function for Cables on Real-Life Bridges
(Page 3 – 6)

Creating a Parabola from a Created Function
(Page 6 – 8)

Problem Solving for Real-Life Examples
(Page 8 – 9)

Creating a Function with Only Variables
(Page 9 – 10)

Creating Function for Golden Gate Bridge
(Page 10 – 11)

Calculating Angles Using Golden Gate Bridge
(Page 11 – 12)

Difference Between Parabola and Catenary
(Page 13)

Conclusion on Parabola Investigation
(Page 14)

Sources
Golden Gate Bridge Measurements:
http://www
...
com/applications-integration/11-arc-length-curve
...
We are mainly going to be using the turning point formula since we are dealing with
parabolas
...


Investigation:

Fig
...
1 is a graph we have been given and we need to find the quadratic function of the parabola,
and since it is a parabola, we will be using the turning point formula:
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘

The y-intercept is the point where the parabola cuts the y-axis, and reading from Fig
...
We also know the coordinate of the vertex, since the vertex is
the point at which the parabola “turns around”, which is the minimum or maximum point of
the parabola
...

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 => 𝑦 = 𝑎(𝑥 − 1)2 + 0

2

Now that we know these values, we can calculate a by inputting the values into the equation
...
2

Fig
...
2, we can see that the y-intercept is (0,2)
...
Now we
plug in the values
...

Since we need a point on the graph, and we already have two, we will simply be using the yintercept point, which is (0,2):

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 => 2 = 𝑎(0 − 5)2 + 0 = 2 = 25𝑎 =>
By this, we know that 𝐚 =

𝒚=

𝟐
𝟐𝟓

2
= 𝑎
25

, and therefore the final equation is:

𝟐
(𝒙 − 𝟓) + 𝟎
𝟐𝟓

Fig
...

Fig
...
We also know the
coordinates of the vertex, since we know that the x-coordinate of the vertex is in the middle of
the main span, and half of the main span is:

4

10m
= 5m
2
As stated, the cable sags to a height of 2 meters above the deck, which is the y-coordinate of
the vertex
...
Now we plug in the values:
3 = 𝑎(𝑥 − ℎ)2 + 𝑘 => 3 = 𝑎(𝑥 − 5)2 + 2

Now that we know these values, we can calculate a by inputting the values into the equation
...
We are going to be using the left pylon and
going 2 meters inward from that one
...
We also know the
coordinates of the vertex, since we calculated them before, where they were v(5,2)
...
36
25
25

By this we know that y = 2
...
36 meters
...
4, where the black lines represent the area we
used to calculate the above calculations:

5

Fig
...
We have
been given these points:





The pylons are 255 meters above the water
The main span is 1625 meters
The road deck is 70 meters above the water
The height of the cable at its lowest sag is negligible, which means that the cable goes all
the way down to the main deck

Before we can model this, we need to find the function
...
We know the
height of the pylons from the water and the height of the main deck from the water, so we just
subtract the two, and we have the height of the pylons from the main deck:
255m − 70m = 185m

6

Therefore, the y-intercept is (0,185)
...
The x-coordinate is simply half of the main span, since the cable touches the main
deck in the middle
...
5m
2

Therefore v(812
...
Now we plug in the values:
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 => 185 = 𝑎(𝑥 − ℎ)2 + 𝑘 => 185 = 𝑎(𝑥 − 812
...

Since we need a point on the graph, and we already have two, we will simply be using the yintercept point, which is (0,185):

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 => 185 = 𝑎(0 − 812
...
25𝑎 =>
185
= 𝑎 = 2
...
80 × 10−4 (3 𝑠
...
25

And the final equation becomes:

𝒚 = 𝟐
...
The diagram turned out as followed on Fig
...
5

Now a person is crawling along this cable
...
We need to how high the person is above the main deck
...
And since we know his xcoordinate, we can easily find his y-coordinate:
𝑦 = 2
...
5)2 + 0 => 𝑦 = 2
...
5)2 + 0 = 105
...
)

Therefore, his y-coordinate is 105 meters and that is also his height above the main deck
...
So that it does not fly into the cable, we need to
calculate how many meters away from the pylons the pilot should fly
...

We need to know the pilot’s y-coordinate from the main deck and from sea level
...
80 × 10−4 (𝑥 − 812
...
80 × 10−4 (𝑥 − 812
...
5)2 => √
+ 812
...
28m
−4
2
...
80 × 10−4

This is the range at which in the pilot can fly between the pylons, and to find the distance the
pilot needs to fly from the pylons, to be in a safe distance, we need to subtract this value by the
length of the main span:
1625m − 1439
...
72m = 186m (3 s
...
)
Therefore, the pilot needs to fly 186 meters from each pylon to be in a safe distance and to get
a good view
...
We
have been given these variables:




It’s a suspension bridge with a center span of l meters
It’s pylons are of height h meters above water
The road deck is c meters above water

We start with the turning point formula:
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
To get the y-intercept we need to subtract the height of the road deck above the water from
the height of the pylons above the water
...
The x-coordinate of the
𝟏

vertex is half of the center span, so it becomes 𝟐 𝐥
...
Now we put in these new variables:
1 2
ℎ − 𝑐 = 𝑎 (𝑥 − 𝑙) + 0
2

9

We can rearrange this to get a:
1 2
ℎ − 𝑐 = 𝑎 (𝑥 − 𝑙) + 0 => 𝑎 =
2

ℎ− 𝑐
1 2
(𝑥 − 2 𝑙)

Now we put this form of a into our first function:

𝑦=

1 2
𝑙) + 𝑘
2 × (𝑥 −
2

ℎ− 𝑐
1
(𝑥 − 2 𝑙)

So this becomes the final function
...
We are going to be finding the
function of the bridge spans of the Golden Gate Bridge
...
6

Now with these two values we have in Fig
...
7109 × 10−4 = 3
...
f
...
71 × 10−4 × (𝑥 − 640)2 + 0
Fig
...


Fig
...
Fig
...


Fig
...
The
adjacent is the side from the angle that leads down to the right angle, in this case 152 meters
...
639 𝑜 = 76
...
)
152
So the angle between one of the pylons and the cable is 76
...

Now we need to find the length of the cable from one of the pylons to the other pylon on the
Golden Gate Bridge
...
71 × 10−4 )2 ) × 𝑋 2 ) × 𝑑𝑋 = 1326
...
)

Therefore, the distance of the cable from one pylon to the other is 1452 meters
...
For example, a chain attached on one
pole and the other end attached to another pole
...
For example a
suspension bridge, where the cable is hanging from the two pylons with the road deck below
...

The cable on the Golden Gate Bridge that makes the side span connect the pylon to the
stiffening girder is a catenary, since it is simply hanging by itself, and doesn’t have equal load
like written above
...
9 shows a catenary, and it is a catenary because it does not support anything
...

Fig
...


Fig
...
10

13

In this investigation we have gone through parabolas and how to use the turning point formula
to find the function of a parabola or use it to create a parabola from a function
...
We have put this knowledge to use in real-life examples on suspension
bridges where we created a function for suspension bridges and created our own bridges out of
a function
...

When we had learned the basics, we created a function out of no values, but variables, which
we could put to use later in real bridges and actual measurements, to create a function for a
real bridge, in this case the Golden Gate Bridge
...
We then moved onto calculating the angle between the pylon and the
cable on the Golden Gate Bridge using tangent

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