Two control towers are located at points Q(500, 0) and R(500, 0), on a straight shore where the x-axis runs through (all distances are in meters). At the same moment, both towers sent a radio signal to a ship out at sea, each traveling at 300 m/µs. The ship received the signal from Q 3 µs (microseconds)before the message from R. Find the equation of the curve containing the possible location of the ship.

9514 1404 393

  x^2/81 -y^2/19 = 2500

Step-by-step explanation:

The time difference between signals is 3 µs, so the distance difference is ...

  (300 m/µs)(3 µs) = 900 m

If we assume the coordinates of Q are (-500, 0), then the distances from point X to the control towers are ...

  XQ = √((x +500)^2 +y^2)

  XR = √((x -500)^2 +y^2)

We want the difference in distances to be 900, so we have the equation ...

  |XQ -XR| = |√((x +500)^2 +y^2) -√((x -500)^2 +y^2)| = 900

Squaring both sides gives ...

  (x +500)^2 +2(x +500)y +y^2 -2√(((x +500)^2 +y^2)((x -500)^2 +y^2)) +((x -500)^2 +y^2) = 810000

Separating the root from the rest of the equation, squaring again, and simplifying the rather messy expression, we can arrive at the equation ...

  x^2/81 -y^2/19 = 2500 . . . . . a hyperbola opening horizontally

2 over 5

step-by-step explanation:

the 2: 3 ratio is the length of the ac segment to the cb segment total length. this implies that the ratio between segment ac length and segment ab length is 2: 5. point c is 2/5 from a to b, so the required fraction is 2/5.

u = -(6/7)w

step-by-step explanation:

place all values with 'u' on one side of the equation.

4u + 8w = - 3u + 2w

4u + 3u = 2w - 8w

further simplify the equation by solving the addition and subtraction.

7u = - 6w

make 'u' the independent variable by placing its co-efficient on the right-hand side.

u = - (6/7)w