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Let's break down the problem:

**Given:**
- Two control towers are located at \( Q(-500, 0) \) and \( R(500, 0) \) along the shore (x-axis).
- A ship out at sea receives a radio signal sent simultaneously from both towers, with each signal traveling at \( 300 \, \text{m/μs} \).
- The ship receives the signal from \( Q \) 3 μs earlier than from \( R \).

### Part (a) - Equation of the curve containing the possible location of the ship:
Let the position of the ship be \( (x, y) \).

- The distance from the ship to tower \( Q \) is:
  \[
  d_Q = \sqrt{(x + 500)^2 + y^2}
  \]
  
- The distance from the ship to tower \( R \) is:
  \[
  d_R = \sqrt{(x - 500)^2 + y^2}
  \]
  
- The difference in the time it takes for the signals to reach the ship from \( Q \) and \( R \) is 3 μs.
  
  Since the signal speed is \( 300 \, \text{m/μs} \), the difference in distance is:
  \[
  d_Q - d_R = 300 \times 3 = 900 \, \text{m}
  \]
  
Thus, the equation becomes:
\[
\sqrt{(x + 500)^2 + y^2} - \sqrt{(x - 500)^2 + y^2} = 900
\]
This is the equation of the curve that represents the possible locations of the ship.

### Part (b) - Coordinates when \( y = 200 \):
To find the exact location when the ship is 200 m from the shore (i.e., \( y = 200 \)), we substitute \( y = 200 \) into the equation derived in part (a) and solve for \( x \).

Now, let’s compute this step-by-step.

### Solving Part (b):
1. Substituting \( y = 200 \) into the equation:
   \[
   \sqrt{(x + 500)^2 + 200^2} - \sqrt{(x - 500)^2 + 200^2} = 900
   \]
   
2. Expand the squares and solve the equation for \( x \).

Let me calculate this for you.The solution for \( x \), when \( y = 200 \), is:

\[
x = \frac{450 \sqrt{665}}{19} \approx 408.65
\]

Thus, the coordinates of the ship when it is 200 m from the shore are approximately \( (408.65, 200) \).

Would you like further details or have any other questions?

Here are 5 follow-up questions related to the problem:
1. How would the equation change if the difference in time were 5 μs instead of 3 μs?
2. Can you derive the general form of the equation for the curve given different tower positions?
3. What if the signal speed were slower, say 250 m/μs? How would that impact the location of the ship?
4. What geometric shape does the curve of possible ship locations resemble?
5. How would the location of the ship change if the towers were not aligned along the x-axis?

**Tip:** Problems like these often involve using the concept of a hyperbola, where the difference in distances to two fixed points (foci) is constant.

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 300m/μs. The ship received the signal from Q 3μs (microseconds) before the message from R. (a) Find the equation of the curve containing the possible location of the ship. (b) Bonus question: Find the coordinates (rounded off to two decimal places) of the ship if it is 200 m from the shore (y = 200).

Explore how to find the possible location of a ship at sea using hyperbolic geometry. This problem involves calculating the difference in distances between two control towers, applying the distance formula, and solving for coordinates when the ship is 200 meters from the shore.

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