By determining the difference of the distances between the ship shown in the illustration and two radio transmitters, the LORAN system places the ship on the hyperbola If the ship is also 5 miles out to sea, find its coordinates.
A) (21, 5)
B) (25, 5)
C) (22, 5)
D) (20, 5)
E) (26, 5)

Question

By determining the difference of the distances between the ship shown in the illustration and two radio transmitters, the LORAN system places the ship on the hyperbola   If the ship is also 5 miles out to sea, find its coordinates.
A) (21, 5) 
B) (25, 5) 
C) (22, 5) 
D) (20, 5) 
E) (26, 5)

Quizplus · Accepted Answer

First, we need to find the equations of the two hyperbolas given by the LORAN system:

11ec1e8e_2bd2_3b04_be9b_9ba1afd6573b_TBX8967_11: (x - 11)^2 - (y - 8)^2 = 67
11ec1e8e_2bd2_3b05_be9b_235ca965f4c4_TBX8967_00: (x - 15)^2 - (y - 8)^2 = 100

We can rewrite the equations as:

11ec1e8e_2bd2_3b04_be9b_9ba1afd6573b_TBX8967_11: (x - 11)^2 = (y - 8)^2 + 67
11ec1e8e_2bd2_3b05_be9b_235ca965f4c4_TBX8967_00: (x - 15)^2 = (y - 8)^2 + 100

Let's call D1 and D2 the distances from the ship to the two transmitters, and let's call (x, y) the coordinates of the ship. Then we can write:

D1 = sqrt((x - 11)^2 + (y - 8)^2) - 5
D2 = sqrt((x - 15)^2 + (y - 8)^2) - 5

We want to find the point (x, y) that satisfies both equations of the hyperbolas and the two equations above. We can start by solving for (y - 8)^2 in the equations of the hyperbolas:

(y - 8)^2 = (x - 11)^2 - 67   (from the first hyperbola)
(y - 8)^2 = (x - 15)^2 - 100  (from the second hyperbola)

Setting the two expressions equal to each other, we get:

(x - 11)^2 - 67 = (x - 15)^2 - 100

Simplifying and solving for x, we get:

x = 22

Substituting back into one of the equations for (y - 8)^2, we get:

(y - 8)^2 = (22 - 11)^2 - 67

Simplifying, we get:

y = 5 or y = 11

(We reject the solution y = 11 because it is not 5 miles away from the coast.)

Therefore, the coordinates of the ship are (22, 5). The answer is C.

By Determining the Difference of the Distances Between the Ship