To solve this problem, we need to determine how far from the vertical beam the lower end of the support beam should be placed along the horizontal floor. We can map this situation to a right-angled triangle where:
- The vertical beam represents the height (opposite side) of the triangle.
- The segment of the support beam meeting the floor forms the adjacent side.
- The angle given is the angle of elevation (28 degrees).
Here’s how to solve it step-by-step:
1. Identify the known values:
- Angle of elevation (A): [tex]\( 28^\circ \)[/tex]
- Height (opposite length, [tex]\( C \)[/tex]): 1.6 meters
2. We need to find the length of the adjacent side (let’s call it [tex]\( L \)[/tex]), which is the horizontal distance from the vertical beam to the lower end of the support beam.
3. Since we are dealing with a right triangle, we can use trigonometric ratios. Specifically, the tangent (tan) of the angle [tex]\( A \)[/tex] is defined as:
[tex]\[
\tan(A) = \frac{{\text{opposite}}}{{\text{adjacent}}}
\][/tex]
For our triangle:
[tex]\[
\tan(28^\circ) = \frac{1.6}{L}
\][/tex]
4. Rearrange the equation to solve for [tex]\( L \)[/tex]:
[tex]\[
L = \frac{1.6}{\tan(28^\circ)}
\][/tex]
5. Using the tangent value for [tex]\( 28^\circ \)[/tex]:
[tex]\[
\tan(28^\circ) \approx 0.5317
\][/tex]
6. Calculate the adjacent length [tex]\( L \)[/tex]:
[tex]\[
L = \frac{1.6}{0.5317} \approx 3.009\ \text{meters}
\][/tex]
Given the options, the closest value to 3.009 meters is 3.0 meters.
Therefore, the approximate horizontal distance from the vertical beam to the lower end of the support beam should be:
- 3.0 meters.
