In the Adirondack Mountains near Tupper Lake, New York, forest fire observers used fire observation towers for over 70 years. The people who staffed the towers reported forest fires and smoke sightings. As many as 57 towers were used by the observers; however, in the early 1980s, the State of New York determined that the towers were no longer needed since aerial observations had become commonplace.
Suppose a forest ranger in one of the observation towers in the Adirondacks sights a fire 32° east of north while a ranger in a tower 13 miles due east of the first ranger sights the same fire at 49° west of north. How far is the fire from each ranger? (Round your answers to two decimal places.)
To tackle this problem, we can use trigonometry and create a diagram to visualize the situation.
Let's start by drawing the two observation towers, one labeled as Tower 1 and the other as Tower 2. We'll also label the sightings of the fire as point F.
Now, let's mark the angles given in the problem. From Tower 1, the angle is 32° east of north, which means we measure the angle clockwise from the north direction. From Tower 2, the angle is 49° west of north, which means we measure the angle counterclockwise from the north direction.
Next, we connect the two observation towers with a straight line segment labeled as d (representing the distance between the towers). We now have a triangle formed by the two towers and the location of the fire.
To find the distance of the fire from each ranger, we need to determine the lengths of the two sides of the triangle formed by the towers and the fire.
Let's assume the distance from Tower 1 to the fire is labeled as x, and the distance from Tower 2 to the fire is labeled as y.
Now, we can use trigonometry to solve for x and y. We'll use the concept of trigonometric ratios for right triangles.
From Tower 1:
tan(32°) = x / d
From Tower 2:
tan(49°) = y / d
We have two equations with two unknowns (x and y), so we can solve for them simultaneously.
To find x:
x = d * tan(32°)
To find y:
y = d * tan(49°)
Now, we need to find the value of d. In the problem, it mentions that the distance between the towers is 13 miles. Therefore, d = 13 miles.
Plugging in the given values, we have:
x = 13 miles * tan(32°)
y = 13 miles * tan(49°)
Evaluating these expressions using a calculator, we find:
x ≈ 8.49 miles
y ≈ 16.96 miles
So, the fire is approximately 8.49 miles away from Tower 1 and 16.96 miles away from Tower 2.
the base of the triangle is the east-west line between the rangers
the angle at the 1st ranger is ... 90º - 32º
the angle at the 2nd ranger is ... 90º - 49º
the 3rd angle is between the ranger stations (from the fire)
... this distance is 13 mi
use the law of sines to find the other two distances (rangers to fire)