Trig problem

Question

From a point on the floor the angle of elevation to the top of a door is 47 degrees, while the angle of elevation to the ceiling above the door is 59 degrees. If the ceiling is 9.8 feet above the floor, what is the vertical dimension of the door?

Answer 1

draw a right triangle with the hypotenuse on the left and the vertical door on the right

let d=dimension of the door

let l=distance from the point on the floor to the door

tan(47º)=d/l

this implies l=d/tan(47º)

tan(47º)=1.0723687

l=d/1.0723687

now extend the vertical door up to the ceiling and form another right triangle with the height to the ceiling being 9.8 feet

and the angle of elevation being 59º

tan(59º)=9.8/l where l=d/1.0723687

tan(59º)=1.6642794

1.6642794=9.8/(d/1.0723687)

1.6642794=(9.8)(1.0723687)/d

1.6642794=10.50921326/d

d=10.50921326/1.6642794

d=6.314572697 feet

the dimension of the door is 6.3 feet