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Two stations, located a great distance apart, simultaneously transmit radio pulses to ships at sea. Since a ship is usually closer to one station than the other, the ship receives these pulses at slightly different times. By measuring the time differential and by knowing the speed of the radio waves, a ship can be located on the hyperbola whose foci are the positions of the two stations. Suppose stations A and B are located 400 miles apart along a straight shore, with A due west of B. A ship approaching the shore receives radio pulses from the stations and is able to determine that it is 100 miles farther from station A than it is from Station B.

a. Find the equation of the hyperbola on which the ship is located? (What value does the difference of the distance from the ship to each station give you? (2a is supposed to be the difference)

b. Find the exact coordinates of the ship if it is 60 miles from shore.

2007-04-10 13:03:02 · 2 answers · asked by morgulis2003 3 in Science & Mathematics Mathematics

2 answers

Let the shoreline be the x-axis, and suppose A is at (0, 0) and B at (400, 0). If the ship is at (x, y) then the distance from the ship to A is √(x^2 + y^2) and the distance to B is √((x-400)^2 + y^2), so we have
√(x^2 + y^2) - 100 = √((x-400)^2 + y^2)
<=> x^2 + y^2 + 10000 - 200√(x^2 + y^2) = x^2 - 800x + 160000 + y^2
<=> 200√(x^2 + y^2) - 800x + 150000 = 0
<=> √(x^2 + y^2) - 4x + 750 = 0
<=> x^2 + y^2 = 16x^2 - 6000x + 562500
<=> y^2 = 15x^2 - 6000x + 562500

y = 60 => 15x^2 - 6000x + 562500 - 3600 = 0
=> x^2 - 400x + 37260 = 0
=> (x - 200)^2 = 2740
=> x = 200 ± 2√685.

2007-04-10 19:59:16 · answer #1 · answered by Scarlet Manuka 7 · 0 0

Let the two stations be located at the foci:

A(0,0) and B(400,0).

The foci are at a distance 2c apart.

2c = 400
c = 200

The center (h,k) of the ellipse is the midpoint of the foci.

(h,k) = (200,0)

The ship is 100 miles farther from A than B.

2a = 100
a = 50
a² = 2,500

Now solve for b².

b² = c² - a² = 200² - 50² = 40,000 - 2,500 = 37,500

The equation of the hyperbola is:

(x - h)²/a² - (y - k)²/b² = 1

(x - 200)²/2,500 - y²/37,500 = 1
__________________

b. Find the exact coordinates of the ship if it is 60 miles from shore.

In other words solve for x when y = 60.

(x - 200)²/2,500 - y²/37,500 = 1

Multiply thru by 37,500 to clear the denominators.

15(x - 200)² - y² = 37,500

15(x - 200)² = y² + 37,500 = 60² + 37,500 = 3,600 + 37,500

15(x - 200)² = 41,100

(x - 200)² = 2,740

x² - 400x + 40,000 = 2,740

x² - 400x + 37,260 = 0

x = [400 ± √(400² - 4*1*37,260)] / (2*1)

x = (400 ± √10,960) / 2 = (400 + 4√685) / 2

x = 200 ± 2√685 ≈ 252.34501
Since the ship is closer to B than A, the solution with the minus sign is rejected.

x = 200 + 2√685 ≈ 252.34501

The location of the ship is (x,y) = (200 + 2√685, 60).

2007-04-13 18:23:16 · answer #2 · answered by Northstar 7 · 0 0

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