Thread: Hyperbola Problem

Listening post are at A, B, and C. Point A is 2000ft north of point B, and point C is 2000ft east of B. The sound of a gun reaches A and B simultaneously one second after it reaches C. Show that the coordinates of a gun's position are approximately (860,1000), where the x - axis passes through B and C and the origin is midway between B and C. Assume that sound travels 1100ft/sec

Originally Posted by ^_^Engineer_Adam^_^

Listening post are at A, B, and C. Point A is 2000ft north of point B, and point C is 2000ft east of B. The sound of a gun reaches A and B simultaneously one second after it reaches C. Show that the coordinates of a gun's position are approximately (860,1000), where the x - axis passes through B and C and the origin is midway between B and C. Assume that sound travels 1100ft/sec

The coordinates of the points A, B and C are: (-1000, 2000), (-1000,0), (1000,0).

Let the gun's position be (x,y), the we are effecticly told that (as the
distances of the gun from A and from B are equal):

(-1000-x)^2 + (2000-y)^2 = (-1000-x)^2 + y^2.

Which fixes y=1000.

Also we are told that as the sound reaches B 1 second after it reaches C:

sqrt[(-1000-x)^2 + y^2] = sqrt[(1000-x)^2 + y^2] + 1100

This is a curve with a single branch and so if y=1000, has a single root,
and this can be found numericaly to be at x=858.0.

(you can substitute the suggested x=860 y=1000 into the above equation
and show that the two sides are approximatly equal)

RonL

Originally Posted by ^_^Engineer_Adam^_^

Listening post are at A, B, and C. Point A is 2000ft north of point B, and point C is 2000ft east of B. The sound of a gun reaches A and B simultaneously one second after it reaches C. Show that the coordinates of a gun's position are approximately (860,1000), where the x - axis passes through B and C and the origin is midway between B and C. Assume that sound travels 1100ft/sec

HI,

CaptainBlack types twice as fast as I can do, so I don't reply to your problem.

If you like to get some further information see here: LORAN - Wikipedia, the free encyclopedia

EB

Hello, ^_^Engineer_Adam^_^!

I have no idea why they placed the origin that way . . . very inconsiderate!

Listening post are at A, B, and C.
Point A is 2000ft north of point B, and point C is 2000ft east of B.
The sound of a gun reaches A and B simultaneously one second after it reaches C.
Show that the coordinates of a gun's position are approximately (1860,1000),
where the origin is at B. .Assume that sound travels 1100ft/sec

Code:

    A *
      |*
      | *
      |  *
      |   * d+1100
      |    *
 2000 |     *
      |      *  G
      |       *(x,y)
      |     *     *    d+1100
      |   *d          *
      | *                 *
      * - - - - - - - - - - - *
      B         2000          C

The gun is at G(x,y).
Let GB = d. .Then GA = GC = d + 1100

Since G is equidistant from A and C, it lies on the 45°-line: x = y

The distance from G(x,x) to B(0,0) is: .√(2x²) .= .d

The distance from G(x,x) to C(2000,0) is: .√[(x - 2000)² + x²] .= .d + 1100


And we have this system of equations:

. . . . . . . . . . . 2x² .= .d²

. . (x - 2000)² + x² .= .(d + 1100)²


Good luck!