SOLUTION: A pilot approaching a 3000 meter runway finds that the angles of depression of the ends of the runway are 14 and 20. How far is the plane from the nearer end of the runway?

Question 958729: A pilot approaching a 3000 meter runway finds that the angles of depression of the ends of the runway are 14 and 20. How far is the plane from the nearer end of the runway?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the angle of depression of 20 degrees takes him to the near end of the runway.

that makes an angle of 70 degrees to the ground as shown in the diagram.

the height of the airplane is equal to h.

the ground distance from the plane to the start of the runway is x.

that's the top diagram in the picture.

the angle of depression of 14 degrees takes him to the far end of the runway.

that makes an angle of 76 degrees to the ground as shown in the diagram.

the height of the airplane is equal to h as well.

the ground distance from the plane to the end of the runway is x + 3000.

in the top diagram, tan(70) = x/h.

solve for h to get h = x / tan(70).

in the bottom diagram, tan(76) = (x + 3000) / h

solve for h to get h = (x + 3000) / tan(76)

since both equations are equal to h, you can set them equal to each other to get:

x / tan(70) = (x + 3000) / tan(76)

cross multiply to get x * tan(76) = (x + 3000) * tan(70)

distribute the multiplication on the right side of the equation to get:

x * tan(76) = x * tan(70) + 3000 * tan(70)

subtract x * tan(70) from both sides of the equation to get:

x * tan(76) - x * tan(70) = 3000 * tan(70)

factor out the x to get:

x * (tan(76) - tan(70)) = 3000 * tan(70)

divide both sides of the equation by (tan(76) - tan(70)) to get:

x = (3000 * tan(70)) / (tan(76) - tan(70))

solve for x to get:

x = 6524.506713 meters.

that's the ground distance from the plane to the start of the runway.

you can also solve for height to get height = x / tan(70) which becomes:

height = 6524.506713 / tan(70) = 2374.7267237.

diagrams are shown below:

$$$

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