Question

A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. Find the height h of the roof. Round your answer to the nearest tenth.

Answer 1

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

$BD^{2}=BC^{2}+CD^{2}\\\\BD=\sqrt{BC^{2}+CD^{2}}\\\\=\sqrt{10^{2}+5^{2}}\\\\=\sqrt{125}$

Compute the measure of angle x as follows:

$tan\ x=\frac{BC}{CD}\\\\tan\ x=\frac{10}{5}\\\\tan\ x=2\\\\x=tan^{-1}2\\\\x=63.4$

Now consider the triangle ABD.

The hypotenuse AD is the height of the roof.

Determine the measure of side AD as follows:

$cos\ x=\frac{BD}{AD}\\\\cos\ 63.4=\frac{\sqrt{125}}{AD}\\\\0.4478=\frac{\sqrt{125}}{AD}\\\\AD=\frac{\sqrt{125}}{0.4478}\\\\AD=24.967\\\\AD\approx 25$

Thus, the height of the roof is 25 ft.