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### Parabola Problems

```
Date: 02/23/97 at 19:00:06
From: Cindy Smith
Subject: Re:  Parabola Problems

A culvert is shaped like a parabola, 18 centimeters across the top
and 12 centimeters deep.  How wide is the culvert 8 centimeters from
the top?

I have tried setting it up as a complete the square expression, but I
haven't been able to get the correct answer of 6sqrt(3).

Cindy Smith
```

```
Date: 03/08/97 at 02:23:22
From: Doctor Luis
Subject: Re:  Parabola Problems

Cindy,

Here's a modest diagram of a parabola:

^
y  |
(-9,12)*      |      * (9,12)  [ 18 cm across the top ]
|
*     |     *
|
*    |    *
|
*   |   *
|
* | *
---------------*----------------------> x
|
|

The equation for the parabola, in this case, is:

y-k = a*(x-h)^2
y = a * x^2

Since you know that the point (9,12) is on the parabola:

12 = a * 9^2
12 = 81a
a = 12/81
= 4/27

Now the equation is: y = (4/27)*x^2

If you're 8 cm from the TOP (I'm considering the top to be y = 12),
then you're at (x,12-8) = (x,4) [we still haven't found x for this
point]. Therefore:

4 = (4/27) * x^2
x^2 = 4*27/4
= 27
x = sqrt(27) = 3 * sqrt(3)

Since x represents the half-width of the parabola, the width at 8 cm
from the top is: 2x = 6 * sqrt(3)

-Doctor Luis,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

```
Associated Topics:
High School Equations, Graphs, Translations

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