Solve the problem.
-Two factories blow their whistles at exactly the same time. If a man hears the two blasts exactly $6.3$ seconds and $5.2$ seconds after they are blown and the angle between his lines of sight to the two factories is $62.6 ^ { \circ }$, how far apart are the factories? Give your result to the nearest meter. (Use the fact that sound travels at $344 \mathrm {~m} / \mathrm { sec }$ )
A) 2472 meters
B) 3111 meters
C) 2080 meters
D) 3386 meters

Question

Quizplus · Accepted Answer

The distance the sound travels from each factory to the man can be found by multiplying the time it takes for the sound to reach him by the speed of sound. For the first factory, it's $6.3 \times 344 = 2167.2$ meters, and for the second factory, it's $5.2 \times 344 = 1788.8$ meters. These distances form two sides of a triangle, with the line connecting the factories as the third side. Using the Law of Cosines, $c^2 = a^2 + b^2 - 2ab\cos(C)$, where $a = 2167.2$, $b = 1788.8$, and $C = 62.6^\circ$, we can solve for $c$, the distance between the factories. Plugging in the values, we get $c \approx 2080$ meters, rounding to the nearest meter.

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