Question

a support beam needs to be placed at a 28° angle of elevation so that the top meets a vertical beam 1.6 meters above the horizontal floor. the vertical beam meets the floor at a 90° angle. approximately how far from the vertical beam should the lower end of the support beam be placed along the horizontal floor? law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

1. First, identify the trig - function to use:

• We have a right - triangle where the height of the vertical beam (opposite side with respect to the given angle) is \(y = 1.6\) meters and the angle of elevation is \(\theta=28^{\circ}\), and we want to find the distance \(x\) (adjacent side with respect to the given angle) along the horizontal floor.
• The trigonometric function that relates the opposite side and the adjacent side of a right - triangle is the tangent function. The formula for the tangent of an angle \(\theta\) in a right - triangle is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

1. Then, substitute the known values into the formula:

• Given \(\theta = 28^{\circ}\) and the opposite side \(y = 1.6\) meters, and we want to find the adjacent side \(x\). So, \(\tan\theta=\tan(28^{\circ})=\frac{1.6}{x}\).
• We know that \(\tan(28^{\circ})\approx0.5317\). Then, \(x=\frac{1.6}{\tan(28^{\circ})}\).
• Substituting the value of \(\tan(28^{\circ})\), we get \(x=\frac{1.6}{0.5317}\approx3.0\) meters.

Answer:

3.0 meters