Two planes approaching O'Hare from opposite directions have angles of 17.78° and 12.65° to the tower. Their distance to the tower is 8.13 and 11.73 miles respectively, making the total distance between the planes roughly 19.86 miles or 104,300 feet.
Planes Approaching O'Hare: Finding the Distance
Two planes, 104 and 217, approach O'Hare Airport from opposite directions at a 2.5-mile altitude. Flight 104 reports a 17°47' angle of depression to the tower, while 217 reports 12°39'. We need to find the distance between the planes.
First, convert the angles to decimal degrees:
* 17°47' = 17.78°
* 12°39' = 12.65°
Imagine a triangle formed by the two planes, the tower, and their respective paths to the ground. The 2.5-mile altitude forms the triangle's base. We know the angles at the base (17.78° and 12.65°) and need to find the length of the opposite side (distance between the planes).
Using the tangent function:
* For plane 104: tan(17.78°) = h/d, where h is the altitude (2.5 miles) and d is the distance to the tower. Solving, we get d ≈ 8.13 miles.
* For plane 217: tan(12.65°) = h/d, solving, we get d ≈ 11.73 miles.
The total distance between the planes is the sum of their distances to the tower:
* Distance between planes = 8.13 miles + 11.73 miles ≈ 19.86 miles.
Converting to feet:
* Distance between planes ≈ 19.86 miles * 5280 feet/mile ≈ 104,300 feet.
Therefore, the approximate distance between the two planes is 104,300 feet or 19.86 miles.