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### Parabolic Equation of an Arch

```
Date: 05/25/98 at 00:36:49
From: Derek Child
Subject: parabolas

For an assignment in my trig class, I was asked to find the equation
of the parabola in the McDonald's golden arches. I have searched high
and low all over the Internet. I don't want a full blown solution,
just some hints as to how I can come up with a solution. I have tried
taking a picture of the "M" and putting it on a x-y axis, but this
doesn't work too well. Any help would be greatly appreciated. Thanks.
```

```
Date: 06/08/98 at 15:41:53
From: Doctor Jeremiah
Subject: Re: parabolas

Dear Derek:

To define a parabola requires three (x,y) coordinates. Measure the
height (I will use the symbol H for the value that you measure) and
width at the widest part (I will use the symbol W for that value) of
one of the parabolas in the "M."

If you draw the parabola like this ...

y

|
|
|
|
|
-W/2         | (0,0)   W/2
--- -+- ----|--------+++++--------|-------- x
|            +    |    +
|          +      |      +
|         +       |       +
H        +        |        +
|                 |
|       +         |         +
|                 |
|                 |
-+-     +         -+- -H      +
(-W/2,-H)              (W/2,-H)

|----------W----------|

... then you have three points: (0,0), (W/2,-H), and (-W/2,-H).

So all you have to do is solve the equation of the parabola with these
three coordinates.

The equation of a parabola is:

y = ax^2 + bx + c

We need to make three equations, because we want to solve for three
variables: a, b and c. So we take our equation and we substitute in
each of our coordinates for x and y, and we end up with three
equations:

0 = a(0)^2 + b(0) + c
-H = a(W/2)^2 + b(W/2) + c
-H = a(-W/2)^2 + b(-W/2) + c

The first obviously makes c = 0.

To solve this problem, we need to find b. We have two equations left
that just happen to be equal to the same thing (-H), so we can equate
the equations and solve for b:

a(W/2)^2 + b(W/2) + c = a(-W/2)^2 + b(-W/2) + c

Now that we know a and b, we can substitute them into the second or
third equation and solve for a.

Now that we have a, b and c, we can change y = ax^2 + bx + c to the
real equation by replacing the letters with the values of those
variables.

If you need more details, mail me back.

-Doctor Jeremiah, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Equations, Graphs, Translations
High School Geometry
High School Practical Geometry

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