Let's call the position of the plane "P", and the positions of clearing A and clearing B "A" and "B", respectively. We want to find the distance from the plane to clearing A.
First, draw a diagram. The angle of depression to clearing A means that the line from the plane to clearing A makes a 32 degree angle with the horizontal. Similarly, the angle of depression to clearing B means that the line from the plane to clearing B makes a 56 degree angle with the horizontal. We know that the distance between clearings A and B is 5.7 km.
Now, we can use trigonometry to find the distance from the plane to clearing A. Let's call this distance "d". We can set up the following equation:
tan(32) = d/x
where x is the horizontal distance from the plane to clearing A. We can rearrange this equation to solve for d:
d = x tan(32)
Similarly, we can set up an equation for the distance from the plane to clearing B:
tan(56) = (x + 5.7) / d
We can rearrange this equation to solve for d:
d = (x + 5.7) / tan(56)
Now we can set the two expressions for d equal to each other and solve for x:
x tan(32) = (x + 5.7) / tan(56)
x tan(32) tan(56) = x + 5.7
x (0.6561) = x + 5.7
0.6561 x = x + 5.7
-0.3439 x = 5.7
x = -5.7 / 0.3439
x ≈ -16.57
Since the distance from the plane to clearing A is a positive distance, we can ignore the negative solution. Therefore, the distance from the plane to clearing A is approximately:
d = x tan(32) ≈ (-16.57) tan(32) ≈ 9.18 km
Rounding this to the nearest hundredth of a kilometer gives:
d ≈ 9.18 km
Therefore, the distance from the plane to clearing A is approximately 9.18 km.
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