Problem 51 - College Algebra

Question

An explosion is recorded by two microphones that are 1 mile apart. Microphone \(M_{1}\) received the sound 2 seconds before microphone \(M_{2} .\) Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones.

Step-by-Step Solution

Answer

The explosion could have taken place at two possible locations either 2200 feet to the right or 3080 feet to the left of microphone \(M_1\).

Step 1: Convert Units

The speed of sound is given as 1100 feet per second, therefore the first step is to convert the distance between the microphones from miles to feet. Since 1 mile is equivalent to 5280 feet, the distance between the microphones is 5280 feet.

Step 2: Calculate Distance

In the 2 seconds that passed between the explosion sound reaching \(M_1\) and \(M_2\), the sound would have traveled a distance of 2200 feet (as distance = speed * time, i.e. 1100 feet/sec * 2 sec = 2200 feet). This means that the explosion could either have happened 2200 feet to the right of \(M_1\) or 5280 - 2200 = 3080 feet to its left.

Step 3: Determine Possible Locations

Therefore, the explosion event could have happened at two possible locations. One location is at 2200 feet to the right of microphone \(M_1\) and the other is at 3080 feet to the left of microphone \(M_1\). If we were to denote the position of the microphones as \(M_1 = 0\) (feet) and \(M_2 = 5280\) (feet), the possible locations of the explosion could be either at 2200 feet or -3080 feet relative to \(M_1\).