Answer:
The question has missing figure, the figure is in the attachment.
The length of wood needed to replace AD is [tex]5.01\ in[/tex].
Step-by-step explanation:
Given;
ABC is a triangle in which AB is of 12 inches in length.
∠DCB = 30° ∠ACD = 15°
Solution,
In ΔABC, BC=12 in
∠ACB = ∠DCB + ∠ACD = [tex]30\°+15\°=45\°[/tex]
tan∠ACB =[tex]\frac{AB}{BC}[/tex]
[tex]tan45\°=\frac{AB}{BC}\\ 1=\frac{AB}{12} \\AB=12\ in[/tex]
Now in ΔDCB,
∠DCB = 30° and BC=12 in
tan∠DCB =[tex]\frac{DB}{BC}[/tex]
[tex]tan30\°=\frac{DB}{12}\\\frac{1}{\sqrt{3} } =\frac{DB}{12} \\DB=\frac{12}{\sqrt{3} } =\frac{12\times\sqrt{3} }{3}=4\sqrt{3}[/tex]
[tex]DB = 4\times1.732=6.928\ in[/tex]
Length of AD = Length of AB - Length of DB
Length of AD= [tex]12-6.928=5.072\approx5.01\ in[/tex]
Thus the length of wood needed to replace AD is [tex]5.01\ in[/tex].
