By what angle should a pole 12 m tall be bent so that the top is 8 m from a point 6 m from the base.

The pole is 12 m high and there is a point A that is 6 m from the base of the pole. The angle that the pole should be bent by so that the top is 8 m from the point A has to be determined.

Let the angle the pole is bent by be x degrees with the horizontal. A triangle is formed by the pole, the ground and a line joining the point A to the top of the pole. If a vertical line of length H is drawn from the top of the pole to the ground, the point A lies between the base of the pole and the point where the vertical touches the ground. Let the distance of this point from the the point A be L. The distance of the point from the base of the pole is 6 + L.

(6 + L)^2 + H^2 = 12^2 and L^2 + H^2 = 8^2

=> 12^2 - (6 + L)^2 = 8^2 -  L^2

=> 144 - 36 - L^2 - 12L = 64 - L^2

=> 144 - 36 - 12L = 64

=> 12L = 44

=> L = 11/3

The angle of the pole is `cos^-1(29/36)` = 36.33 degrees.

The pole has to be bent by an angle equal to 36.33 degrees with the horizontal.