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/