Question

Finding the Pitch of a Roof A carpenter is preparing to put a roof on a garage that is 20 feet by 40 feet by 20 feet. A steel support beam 46 feet in length is positioned in the center of the garage. To support the roof, another beam will be attached to the top of the center beam (see the figure). At what angle of elevation is the new beam? In other words, what is the pitch of the roof?

Short answer

The pitch of the roof is $69°$.

Step-by-step solution

Step 1. Given Information

The dimensions of the garage are $20ft.\times 40ft.\times 20ft.$

Now the steel support beam is placed at the center of the garage whose length is 46ft.

And to support the roof another beam is attached at the top of the center beam.

Step 2. Rough diagram to solve the equations

Now as we can see from the below diagram,

$$\begin{gathered} DA=DE+EA, \\ and, \\ EA=BC=20ft. \\ So,DE=DA-EA=46ft.-20ft.=26ft. \\ Now,In∆CDE, \\ \tan \left(\angle DCE\right)=\frac{DE}{CE}=\frac{26}{10}, \\ So, \\ \angle DCE={\tan }^{-1}\left(\frac{13}{5}\right)\cong 69°. \\ So,theangleofelevationofthenewbeamwillbe69°. \\ OR,Thepitchoftheroofis69°. \end{gathered}$$