Question

Refer to the following diagram for the roof of a house. In the figure, x is the length of a rafter measured from the top of a wall to the top of the roof; θ is the acute angle between a rafter and the horizontal; and h is the vertical distance from the top of the wall to the top of the roof.


Suppose that θ = 32° and h = 37.5 ft.
(a) Determine x. Round the answer to one decimal place.

(b) Find the area of the gable. Round the final answer to one decimal place. [The gable is the triangular region bounded by the rafters and the attic floor.]

Answer 1

anyooooooooooooone?

Answer 2

Since we are given a right triangle we know that
\[\sin(x)=\frac{opp}{hyp}=\frac{37.5}{hyp}=\sin(32)\]
Our hypotenuse being $x$ in this case. From here we find the area of the triangle with
\[A=\frac{1}{2}base \times height\] so we need to find the base. How do we do this? Well
\[\cos(x)=\frac{adj}{hyp}\]
and since we discovered the hypotenuse in the previous part of the problem this should be fairly simple. Hope this helps!