SOLUTION: From a point on the floor the angle of elevation to the top of a door is 48 degrees, while the angle of elevation to the ceiling above the door is 56 degrees. If the ceiling is 10.

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Question 1190989: From a point on the floor the angle of elevation to the top of a door is 48 degrees, while the angle of elevation to the ceiling above the door is 56 degrees. If the ceiling is 10.5 feet above the floor, what is the vertical dimension of the door? Assume the floor and door form a right angle. (Round the the answer to one decimal place.)
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

Here's the diagram

This diagram (and the other one as well) is not to scale.

If you want, it might be beneficial to split the triangles up like so

The goal is to find the value of y, i.e. the length of segment BD.

For now, focus on triangle ABC.
We need to find x
Use the tangent ratio
tan(angle) = opposite/adjacent
tan(C) = AB/BC
tan(56) = 10.5/x
x*tan(56) = 10.5
x = 10.5/tan(56)
x = 7.082339 which is approximate

Now move to triangle BCD. Use that x value and the tangent ratio to find y
tan(angle) = opposite/adjacent
tan(C) = BD/BC
tan(48) = y/x
y = x*tan(48)
y = 7.082339*tan(48)
y = 7.865734 this is also approximate

The last step is to round to the nearest tenth to get 7.9 feet