Question 1172679
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.