Question

1. compute the length of piece ab of the roof truss in figure 35-74.

figure 35-74
details
ab is the top base of a 2.2 meters high trapezoid with a 10.3 meters bottom base and 35.3 and 28.5 degrees side angles.
determining functions of angles in any quadrant
determine the sine, cosine, tangent, cotagent, secant, and cosecant of these angles. for each angle, sketch a right triangle similar to those in figures 36-3, 36-4, and 36-5. label the sides of the triangles + or - . determine the reference angles and functions of the angles. round the answers to 5 significant digits.

1. 115°

Explanation:

Step1: Find the horizontal segments

For the left triangle (angle \(35.3^\circ\)), the horizontal segment length \(x_1\) can be found using \(\cot(35.3^\circ)=\frac{x_1}{2.20}\), so \(x_1 = 2.20\times\cot(35.3^\circ)\). Calculate \(\cot(35.3^\circ)=\frac{1}{\tan(35.3^\circ)}\approx\frac{1}{0.708}\approx1.412\), so \(x_1\approx2.20\times1.412\approx3.106\) m.

For the right triangle (angle \(28.5^\circ\)), the horizontal segment length \(x_2\) can be found using \(\cot(28.5^\circ)=\frac{x_2}{2.20}\), so \(x_2 = 2.20\times\cot(28.5^\circ)\). Calculate \(\cot(28.5^\circ)=\frac{1}{\tan(28.5^\circ)}\approx\frac{1}{0.5417}\approx1.846\), so \(x_2\approx2.20\times1.846\approx4.061\) m.

Step2: Calculate AB length

The bottom base is \(10.30\) m, so the top base \(AB = 10.30-(x_1 + x_2)\). Substitute \(x_1\) and \(x_2\): \(x_1 + x_2\approx3.106 + 4.061 = 7.167\) m. Then \(AB\approx10.30 - 7.167 = 3.133\) m. (Note: More precise calculation of \(\tan(35.3^\circ)\) and \(\tan(28.5^\circ)\) can be done with calculator: \(\tan(35.3^\circ)\approx0.7078\), \(\cot(35.3^\circ)\approx1.4128\), \(x_1 = 2.2\times1.4128\approx3.108\); \(\tan(28.5^\circ)\approx0.5416\), \(\cot(28.5^\circ)\approx1.8463\), \(x_2 = 2.2\times1.8463\approx4.062\); \(x_1 + x_2\approx7.17\); \(AB = 10.3 - 7.17 = 3.13\) m (or more precise: \(10.30 - (3.108 + 4.062)=10.30 - 7.17 = 3.13\) m).

Answer:

The length of \(AB\) is approximately \(\boldsymbol{3.13}\) meters (or more precise value around \(3.13\) - \(3.14\) m depending on calculation precision).