Answer:
The height of the roof is 25 ft.
Step-by-step explanation:
Consider the right angled triangle BCD.
Determine the measure of side BD using the Pythagoras theorem
[tex]BD^{2}=BC^{2}+CD^{2}\\\\BD=\sqrt{BC^{2}+CD^{2}}\\\\=\sqrt{10^{2}+5^{2}}\\\\=\sqrt{125}[/tex]
Compute the measure of angle x as follows:
[tex]tan\ x=\frac{BC}{CD}\\\\tan\ x=\frac{10}{5}\\\\tan\ x=2\\\\x=tan^{-1}2\\\\x=63.4[/tex]
Now consider the triangle ABD.
The hypotenuse AD is the height of the roof.
Determine the measure of side AD as follows:
[tex]cos\ x=\frac{BD}{AD}\\\\cos\ 63.4=\frac{\sqrt{125}}{AD}\\\\0.4478=\frac{\sqrt{125}}{AD}\\\\AD=\frac{\sqrt{125}}{0.4478}\\\\AD=24.967\\\\AD\approx 25[/tex]
Thus, the height of the roof is 25 ft.
