A ranger at forest station A( located at latitude N47 and longitude W116 30') sights smoke from a fire in the direction of S35 36'W. At the same time a ranger at forest station B(located at latitude N48 and longitude W117 45') sights the same fire in the direction of S54 18'E. Locate the fire's longitude and latitude to the nearest minute and it's distance from each of the forest station.
I got the distance from station A and B as 90.409 miles I just don't know where to go next.
which kind of trig are we using?
earth flat at this range :)
In my sketch the position of A and B are as follows:
B is to the "left" of A and "up" from A
the rise is 1° and the run is 1.25°
So the line AB makes an angle of 38.66° with the horizontal.
I then sketched in the position of the fire F and got triangle ABF.
All possible angles at A and B can be found, thus angle F is also known.
So you know all the angles
You said you found AB = 90.409 miles.
But 1° along a great circle = 60 nautical miles
so AB in degrees = √(60^2 + 75^2) = 96.05 nautical miles which is appr 110.55 statute miles.
So I don't see how you got 90.4 miles.
Once that is resolved, use the Sine Law to find the distances between the fire and the stations.
Just a lot of messy conversions to get the coordinates back.
@Steve sine, cosine, and tangent also the law of sines and cosines
@Reiny idk my teacher just told me to use movable-typedotcom to find the distance between the two points
To find the latitude and longitude of the fire, as well as its distance from each forest station, we can use the information provided about the directions and distances. Here's how to proceed:
1. Convert the directions from degrees, minutes to decimal degrees:
- S35 36'W -> (-35.6)°
- S54 18'E -> 54.3°
2. Calculate the latitude and longitude of the fire based on the directions and the known coordinates of the forest stations:
- Starting from Station A (latitude N47, longitude W116 30'), we need to move 35.6° to the south and 30' to the west. This gives us a latitude of N47 - 35.6' and a longitude of W116 30' - 30'.
- Similarly, starting from Station B (latitude N48, longitude W117 45'), we move 54.3° to the south and 18' to the east. This gives us a latitude of N48 - 54.3' and a longitude of W117 45' + 18'.
3. Calculate the distance from each station to the fire. You mentioned that the distance from both stations to the fire is approximately 90.409 miles. To further refine the calculation and obtain more accurate results, we can use more precise formulas, such as the Haversine formula, which considers the Earth's curvature. However, for simplicity, let's assume that the Earth is a perfect sphere and use the simpler formula that assumes a flat surface:
- For Station A, using the latitude and longitude obtained above, calculate the distance to the fire using the Pythagorean theorem: Distance_A = sqrt((latitude_A - latitude_fire)^2 + (longitude_A - longitude_fire)^2).
- For Station B, use the same approach to calculate the distance to the fire: Distance_B = sqrt((latitude_B - latitude_fire)^2 + (longitude_B - longitude_fire)^2).
By following these steps, you can determine the latitude and longitude of the fire to the nearest minute and the distance from each forest station.