Solution

Exercise

Various national and international transmitters continuously transmit radio signals at precisely timed intervals. In the chart shown at left, there are two pairs of such transmitters, F1 and F2, and F3 and F4. Can you explain how the navigator of a ship at sea can find his location by measuring the difference in reception times of each of the two pairs of transmitters?

By measuring the difference in reception times of the pair F1 and F2, we know that the ship is somewhere on the hyperbola that has the transmitters F1 and F2 as foci. Also, if we get the signal from F1 first, that will tell us the ship is closer to F1 than F2. However, the difference in reception times of F1 and F2 alone cannot tell us exactly where the ship is, only that it is somewhere on the hyperbola.

Similarly, the difference in reception times of the pair F3 and F4 can tell us that the ship is somewhere on the hyperbola that has the transmitters F3 and F4 as foci. What point is on both hyperbolas at the same time? Of course, the intersecting point. This is how we can find out the exact location of our ship.

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