SOLUTION: The angles of elevation of the top of a tower measured from two points 103,923 meters apart on the horizontal ground are 30° and 45°. What is the height of the tower?

Algebra ->  Exponents -> SOLUTION: The angles of elevation of the top of a tower measured from two points 103,923 meters apart on the horizontal ground are 30° and 45°. What is the height of the tower?      Log On


   

Question 887403: The angles of elevation of the top of a tower measured from two points 103,923 meters apart on the horizontal ground are 30° and 45°. What is the height of the tower?
Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
ASSUMPTION: The base of the tower and the two points on horizontal ground are collinear.

Draw this description: tower, height y. Two points to the right such that ground to point to top of tower is 45 degrees for the nearer point, and 30 degrees for the farther point. Label distance between the two ground points as 103923 meters. Label the distance from tower base to the nearer point as x.

Nearer point: tan%2845%29=y%2Fx

Farther point: tan%2830%29=y%2F%28x%2B103923%29

Using the tangents for those angles, start with the system,
1=y%2Fx
and
1%2Fsqrt%283%29=y%2F%28x%2B103923%29

y=%28x%2B103923%29%2Fsqrt%283%29
Using x=y and substituting,
y=%28y%2B103923%29%2Fsqrt%283%29
y%2Asqrt%283%29=y%2B103923
y%2Asqrt%283%29-y=103923
y%28sqrt%283%29-1%29=103923
highlight%28y=103923%2F%28sqrt%283%29-1%29%29, and you can finish this computation for the tower height any way you find most comfortable.