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Question 1172679: The following application was developed during World War II. It shows how the properties of hyperbolas can be used in radar and other detection systems.
Two microphones, 1 mile apart, record an explosion. Microphone A receives the sound 2 seconds before microphone B. Where did the explosion occur? (assume sound travels at 1100 feet per second.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely, let's solve this problem using the properties of hyperbolas.
**Understanding the Problem**
* **Hyperbola Property:** A hyperbola is defined as the set of all points where the difference of the distances to two fixed points (foci) is constant.
* **Microphones as Foci:** In this case, the microphones A and B are the foci of the hyperbola.
* **Time Difference:** The 2-second difference in sound arrival times indicates a constant difference in distances from the explosion to the microphones.
* **Sound Speed:** We'll use the speed of sound to convert the time difference into a distance difference.
**Solving the Problem**
1. **Convert Miles to Feet:**
* 1 mile = 5280 feet
2. **Calculate Distance Difference:**
* Distance difference = (time difference) * (speed of sound)
* Distance difference = 2 seconds * 1100 feet/second = 2200 feet
3. **Hyperbola Parameters:**
* The distance between the foci (2c) is 5280 feet.
* Therefore, c = 2640 feet.
* The constant distance difference (2a) is 2200 feet.
* Therefore, a = 1100 feet.
4. **Finding b:**
* We use the relationship c^2 = a^2 + b^2
* b^2 = c^2 - a^2
* b^2 = (2640)^2 - (1100)^2
* b^2 = 6969600 - 1210000
* b^2 = 5759600
* b = sqrt(5759600)
* b = approximately 2400 feet.
5. **Hyperbola Equation:**
* We can set up a coordinate system where the microphones are on the x-axis, with the midpoint between them as the origin.
* The equation of the hyperbola is (x^2 / a^2) - (y^2 / b^2) = 1
* (x^2/1100^2) - (y^2/2400^2) = 1
* (x^2/1210000) - (y^2/5760000) = 1
6. **Determining the Location:**
* The explosion occurred somewhere on the branch of the hyperbola that is closer to microphone A, because A received the sound first.
* Because we do not have any further information, we cannot specify the exact x and y coordinate. We do know the explosion occured on a hyperbola, that has the equation calculated above. We also know that the branch of the hyperbola that is closer to microphone A is the branch that contains the explosion.
**Conclusion**
The explosion occurred on a branch of the hyperbola defined by the equation (x^2 / 1210000) - (y^2 / 5760000) = 1, closer to microphone A. Without additional information, we can't pinpoint the exact coordinates.
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