Question
Atop 50m tower, Jack sees smoke in two areas. One is on a bearing of 40degrees with an angle of depression of 8degrees, and the other on a bearing of 205degrees with and angle of depression of 13degrees. How far apart are the smoke sources?
I'm not sure how bearings work in this question.
I'm not sure how bearings work in this question.
Answers
Reiny
I think we must assume that the tower is in relatively flat plane.
I don't know how well you can sketch 3-D diagrams, but mine has two right-angled triangles.
One showing the first fire, has a base angle of 8° and a height of 50, so the distance of the fire from is:
50/base = tan8, base = 50/tan8°
similarly the 2nd right angled triangle has a base angle of 13° and the distance of the fire from the tower is 50/tan13°
Now the bearing:
the 2nd is 204° and the 1st is 40°
so in now have a triangle with sides 50/tan8 and 50/tan13 with an angle of 204-40 or 165° between them
let the distance between the two fires be x
x^2 = (50/tan8)^2 + (50/tan13)^2 - 2(50/tan8)(50/tan13)cos 165°
I will let you do the button-pushing.
I don't know how well you can sketch 3-D diagrams, but mine has two right-angled triangles.
One showing the first fire, has a base angle of 8° and a height of 50, so the distance of the fire from is:
50/base = tan8, base = 50/tan8°
similarly the 2nd right angled triangle has a base angle of 13° and the distance of the fire from the tower is 50/tan13°
Now the bearing:
the 2nd is 204° and the 1st is 40°
so in now have a triangle with sides 50/tan8 and 50/tan13 with an angle of 204-40 or 165° between them
let the distance between the two fires be x
x^2 = (50/tan8)^2 + (50/tan13)^2 - 2(50/tan8)(50/tan13)cos 165°
I will let you do the button-pushing.