Example 2
Completion requirements
| Example 2 |
Jay is building a shop at his uncle’s acreage. He wants to know how wide the roof trusses are that span the shop. He knows each roof truss is symmetrical, with a length of \(24\) ft on each slope, and an angle of elevation of \(20^\circ\). What is the width, to the nearest tenth of a foot, of each roof truss?
Step 1: Draw a diagram.
The question states that each truss is symmetrical, which indicates that it is an isosceles triangle. Therefore, the two slopes are \(24\) ft and the two angles of elevation are each \(20^\circ\). Form two right triangles by drawing an altitude from the vertex to the base.
The goal of the question is to find the total length of the base, \(d\).
Step 2: Determine \(d_1\).
\(\theta = 20^\circ\)
\({\mathop{\rm adj}\nolimits} = d_1\)
\({\mathop{\rm hyp}\nolimits} = 24\) ft
\(\begin{align}
\cos \theta &= \frac{{{\mathop{\rm adj}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
\cos 20^\circ &= \frac{{d_1 }}{{24}} \\
24\cos 20^\circ &= d_1 \\
22.552... &= d_1 \\
\end{align}\)
Since the two right triangles formed are identical, \(d_2 = 22.552...\) as well.
Step 3: Determine \(d\).
\(\begin{align}
d_1 &= d_2 \\
d &= 2d_1 \\
d &= 2\left( {22.552...} \right) \\
d &= 45.105... \\
d &\doteq 45.1 \\
\end{align}\)
The base of each truss is approximately \(45.1\) ft long.
The question states that each truss is symmetrical, which indicates that it is an isosceles triangle. Therefore, the two slopes are \(24\) ft and the two angles of elevation are each \(20^\circ\). Form two right triangles by drawing an altitude from the vertex to the base.
The goal of the question is to find the total length of the base, \(d\).
Step 2: Determine \(d_1\).
\(\theta = 20^\circ\)
\({\mathop{\rm adj}\nolimits} = d_1\)
\({\mathop{\rm hyp}\nolimits} = 24\) ft
\(\begin{align}
\cos \theta &= \frac{{{\mathop{\rm adj}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
\cos 20^\circ &= \frac{{d_1 }}{{24}} \\
24\cos 20^\circ &= d_1 \\
22.552... &= d_1 \\
\end{align}\)
Since the two right triangles formed are identical, \(d_2 = 22.552...\) as well.
Step 3: Determine \(d\).
\(\begin{align}
d_1 &= d_2 \\
d &= 2d_1 \\
d &= 2\left( {22.552...} \right) \\
d &= 45.105... \\
d &\doteq 45.1 \\
\end{align}\)
The base of each truss is approximately \(45.1\) ft long.