SOLUTION: Two control towers are located at points M(-500,0) and N(500,0),on a straight shore where the x-axis runs through (all distances are in meters). At the same time, both towers sen

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Two control towers are located at points M(-500,0) and N(500,0),on a straight shore where the x-axis runs through (all distances are in meters). At the same time, both towers sen      Log On


   

Question 1204500: Two control towers are located at points M(-500,0) and
N(500,0),on a straight shore where the x-axis runs through (all distances are in meters). At the same time, both towers sent a to a ship out at sea, each traveling at 400m/µs. The ship received the signal from tower N 2µs (microseconds) before the message from M. What is the equation of the curve containing the possible location of the ship? What are the coordinates of the ship if it is 200 m from the shore(y=200)?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!



here we have a hyperbola
x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1

with foci at:M(-500,0) =(-c,0) and N(500,0)=(c,0)

distance between foci is 2a, so to get the distance multiply speed 400%28m%2F%28mu%2As%29%29 by a time 2%28mu%2As%29

2a=2%28mu%2As%29%2A400%28m%2F%28mu%2As%29%29=800m
a=400m
since c=500, we have
c%5E2=a%5E2%2Bb%5E2
500%5E2=400%5E2%2Bb%5E2
b%5E2=500%5E2-400%5E2
b%5E2=90000
b=300

and your hyperbola is:

x%5E2%2F400%5E2-y%5E2%2F300%5E2=1
x%5E2%2F160000+-+y%5E2%2F90000+=+1

What are the coordinates of the ship if it is 200m from the shore(y=200)?
x%5E2%2F160000+-+200%5E2%2F90000+=+1
x%5E2%2F160000+-+4%2F9+=+1
x%5E2%2F160000+=+1%2B4%2F9+
x%5E2%2F160000+=+13%2F9
x%5E2+=+%2813%2F9%29160000
x%5E2+=2080000%2F9
x+=sqrt%282080000%2F9%29
(x+=400sqrt%2813%29%29%2F3 -> exactly
x+=480.74-> approximately


the coordinates of the ship if it is 200m+from the shore are (480.74,200)