Suppose lighthouse A is located at the origin and lighthouse B is located at coordinates (0,6). The captain of a ship has determined that the ship's distance from lighthouse A is 2 and its distance from lighthouse B is 5. What are the possible coordinates for the ship?
Let the coordinates of the ship be P(x,y)
so AP =2 and BP=5
√(x^2+y^2) = 2
x^2+y^2 = 4 , (#1)
√( (x^2 + (y-6)^2 ) = 5
x^2 + y^2 - 12y + 36 = 25
x^2 + y^2 - 12y = -11 , (#2)
#1 - #2 :
12y = 15
y = 15/12 = 5/4 = 1.25
then in #1:
x^2 + 225/144 = 4
x^2 = 39/16
x = ± √39/4
The ship could be at (√39/4 , 5/4) or (-√39/4 , 5/4)
To find the possible coordinates for the ship, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Here, the distance of the ship from lighthouse A is given as 2, and from lighthouse B is given as 5. We can consider these distances as the lengths of the two sides of the right triangle formed by the ship and the two lighthouses. The coordinates of the ship will be the point on this triangle, which is at a distance of 2 from lighthouse A and 5 from lighthouse B.
Let's denote the x-coordinate of the ship as x and the y-coordinate as y. The coordinates of lighthouse A are (0,0) and lighthouse B are given as (0,6). We can form two equations based on the distance formula using the coordinates:
For lighthouse A, the distance formula is:
dA = sqrt((x-0)^2 + (y-0)^2) = 2
For lighthouse B, the distance formula is:
dB = sqrt((x-0)^2 + (y-6)^2) = 5
Squaring both sides of these equations, we get:
(x-0)^2 + (y-0)^2 = 2^2
x^2 + y^2 = 4 -- Equation 1
(x-0)^2 + (y-6)^2 = 5^2
x^2 + (y-6)^2 = 25 -- Equation 2
Now, we have a system of equations. To find the possible coordinates for the ship, we can solve this system.
Subtracting Equation 1 from Equation 2, we get:
(y-6)^2 - y^2 = 25 - 4
y^2 - 12y + 36 - y^2 = 21
-12y = -15
y = 5/4
Substituting the value of y in Equation 1, we get:
x^2 + (5/4)^2 = 4
x^2 + 25/16 = 4
x^2 = 4 - 25/16
x^2 = 64/16 - 25/16
x^2 = 39/16
x = sqrt(39/16)
x = sqrt(39)/4
Therefore, the possible coordinates for the ship are (sqrt(39)/4, 5/4).