# The horizontal distance between two poles is 86 m. The angles of elevation to the top of the poles from a point halfway between them are 14 degrees and 17 degrees. What is the difference in the height of the poles.

It is given that the horizontal distance between the poles is 86 m. A halfway between them is 43 m from each of them.

The angle of elevation of the top of the poles is 14 degrees and 17 degrees.

If the height of the pole with an angle of elevation 14 degrees is taken as H1, we have tan 14 = H1/43,

=> H1 = 43*tan 14

Similarly the height of the pole with the angle of elevation 17 degrees is 43*tan 17

The difference in their height is 43(tan 17 - tan 14) = 2.425 m

The required difference in the height of the poles is 2.425 m

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The distance from each pole to the point in the center is 43m.

(1) The height of the shorter pole is found as follows:

The pole creates a right triangle with acute angles of 14 degrees and 76 degrees. The ratio of the height of the pole to the distance to the center point is equal to the tangent of 14 degrees.

Thus `h/43=tan 14` or `h=43 tan 14 => h ~~ 10.72m`

(2) Similarly, the height of the taller pole is:

`h/43=tan 17` or `h=43tan17 ~~13.15m`

(3) The difference in the heights to the nearest tenth of a meter is:

`13.15-10.72 ~~ 2.4m`