Determine the distance from the campfire to each man, rounded to the nearest foot.

Question

The Colonel spots a campfire at a of bearing N42E from his current position. Sarge, who is positioned 3000 feet due east of the Colonel, reckons the bearing to the fire to be N20W from his current position.

Answer 1

In this scenario, three reference points are given: (a) the colonel’s position, (b) the sarge’s position, and (d) the campfire. With these points, we can draw triangle CSF where angles C (Colonel) and S (Sarge) are base angles separated by 3000 feet (base of the triangle).  A bearing of N42E indicates the bearing that makes an angle of 42 degrees with North towards East. Therefore, angle C measures 48 degrees. A bearing of N20W indicates the bearing that makes an angle of 20 degrees with North towards West. Thus, angle S measures 70 degrees.

The remaining angle F measures 62 degrees (180-48-70 = 62). Now that we know the measures of all three angles and we know the measure of one side, we can use the law of sines to calculate the measures of the remaining sides.

a = c * [sin (A) / sin (62)]  

where c is the base,

a is an unknown side,

and A is the angle opposite of the unknown side.

A = 3000 * [sin 48 / sin 62] will provide the measure of the side opposite of the colonel.

Side opposite of the colonel is 2525 ft. (This side is also the distance between the fire and the sarge.)

A = 3000 * [sin 70 / sin 62] will provide the measure of the side opposite of the sarge.

Side opposite of the sarge is 3193 ft. (This side is also the distance between the fire and the colonel.)