The angle of elevation of the top of a tower is 38 degrees from a point A
due south of it. The angle of elevation of the top of the tower from another point B, due east of the tower is 29degrees. Find the height of the tower if the distance AB is 50 m.
Let the height of the tower be H. From the point A due south of the tower, the top of tower has an angle of elevation is 38 degrees. The distance of the point A from the base of the tower is `H/tan 38` .
Similarly the distance of point B that is due east of the tower is `H/tan 29` .
The distance between A and B is 50 m. The line joining the base of the tower to point A, the line joining the base of the tower to point B and the line joining AB form a right triangle with sides `H/tan 38` , `H/tan 29` and 50 where 50 is the hypotenuse. Using the Pythagorean Theorem gives:
`(H/tan 38)^2 + (H/tan 29)^2 = 50^2`
=> `H^2*(tan^29 + tan^2 38) = 2500*tan^2 38*tan^2 29`
=> `H = sqrt((2500*tan^2 38*tan^2 29)/(tan^29 + tan^2 38))`
=> `H ~~ 22.64 `
The height of the tower is approximately 22.6 m