Question

The Colonel spots a campfire at a bearing N 59∘59∘ E from his current position. Sarge, who is positioned 242 feet due east of the Colonel reckons the bearing to the fire to be N 34∘34∘ W from his current position.

Determine the distance from the campfire to each man, rounded to the nearest foot.
Colonel is about............................ feet away from the fire
Sarge is about............................... feet away from the fire

Answer 1

i. Colonel is about 201 feet away from the fire.

ii. Sarge is about 125 feet away from the fire.

Explanation:

Let the Colonel's location be represented by A, the Sarge's by B and that of campfire by C.

The total angle at the campfire from both the Colonel and Sarge =
`59^(0)` +
`34^(0)`

=
`93^(0)`

Thus,

<CAB =
`90^(0)` -
`59^(0)` =
`31^(0)`

<CBA =
`90^(0)` -
`34^(0)` =
`56^(0)`

Sine rule states;

`(a)/(Sin A)` =
`(b)/(Sin B)` =
`(c)/(Sin C)`

i. Colonel's distance from the campfire (b), can be determined by applying the sine rule;

`(b)/(Sin B)` =
`(c)/(Sin C)`

`(b)/(Sin 56^(0) )` =
`(242)/(Sin 93^(0) )`

`(b)/(0.8290)` =
`(242)/(0.9986)`

cross multiply,

b =
`(0.8290*242)/(0.9986)`

= 200.8993

Colonel is about 201 feet away from the fire.

ii. Sarge's distance from the campfire (a), can be determined by applying the sine rule;

`(a)/(Sin A)` =
`(c)/(Sin C)`

`(a)/(Sin 31^(0) )` =
`(242)/(Sin 93^(0) )`

`(a)/(0.5150)` =
`(242)/(0.9986)`

cross multiply,

a =
`(0.5150*242)/(0.9986)`

= 124.8073

Sarge is about 125 feet away from the fire.