Find the ratio of their height.

Question

A monument and a building were placed on the circumference of a circle of radius "r" ft. on the level ground, the angle of elevation of their tops at the center of the circle would be 30° and 60° respectively.

Answer 1

the monument, the radius, and the hypotenuse from the center of the circle to the top of the monument form a 30-60-90 right triangle

the building, the radius, and the hypotenuse from the center of the circle to the top of the building form another 30-60-90 right triangle

 

in a 30-60-90 right triangle, the longer side is equal to the shorter side times √3

in one triangle the radius is r, the shorter side, and the longer side is r*√3

in the other triangle the radius is r, the longer side, and the shorter side is r/√3

the longer side, r,  is √3*shorter side which makes the shorter side r/√3 so (r/√3)*√3=r

the height of one building is r*√3 and the height of the other building is r/√3

the ratios of the heights is either r√3/(r/√3)=√3√3=3 or 3/1, or...

(r/√3)/r√3)=1/√3√3=1/3