parabolic word problem

[alyren]

An experimental model for a suspension bridge is built in the shape of a parabolic arch. In one section, cable runs from the top of one tower down tot he roadway, just touching it there, and up again to the top of a second tower. the tower are both 9 inches tall and stand 90 inches apart. At some point along the road from the lowest point of the cable, the cables is 1.44 inches above the roadwat. Find the distance between that point and the base of the nearest tower.

please help on this. word problem looks like chinese to me... have no idead what it talking about.

 

[soroban]

Hello, alyren!

An experimental model for a suspension bridge is built in the shape of a parabolic arch.
In one section, cable runs from the top of one tower down tot he roadway, just touching it there,
and up again to the top of a second tower. The towers are both 9 inches tall and stand 90 inches apart.
At some point along the road from the lowest point of the cable, the cables is 1.44 inches above the roadway.
Find the distance between that point and the base of the nearest tower.

. . .             |
      *           |           *(45,9)
      |           |           |
      |*          |          *|
    9 | *         |         * | 9
      |   *       |       *   |
      |      *    |    *  |y  |
  - - * - - - - - * - - - + - * - -
    -45           |   x   P  45

Place the parabola on a coordinate system.
The vertex is at the origin.
The parabola has the form: .\(\displaystyle y \,=\,ax^2\) .[1]

The top of one tower is at (45,9).

Substitute into [1]:
. . . \(\displaystyle 9 \:=\:a(45^2) \quad\Rightarrow\quad a \:=\:\tfrac{1}{225}\)
\(\displaystyle \text{The parabola is: }\:y \:=\:\tfrac{1}{225}x^2\)


\(\displaystyle \text{At point }P,\:y = 1.44\)

\(\displaystyle \text{Hence: }\:1.44 \:=\:\tfrac{1}{225}x^2 \quad\Rightarrow\quad x^2 \,=\,324 \quad\Rightarrow\quad x \,=\,18\)


\(\displaystyle \text{The desired distance is: }\:45-18 \:=\:27\text{ inches.}\)