To solve this problem, let's break it down step-by-step using trigonometric relationships.
Given:
- The angle of elevation at which the support beam is inclined: [tex]\(28^\circ\)[/tex].
- The height of the vertical beam above the floor where the support beam meets: [tex]\(1.6\)[/tex] meters.
We need to find the horizontal distance from the base of the vertical beam to where the lower end of the support beam meets the floor. We will use the tangent function, which relates the angle of elevation, the height of the vertical beam, and the horizontal distance:
[tex]\[
\tan(\theta) = \frac{{\text{{opposite}}}}{{\text{{adjacent}}}}
\][/tex]
Here, [tex]\(\theta\)[/tex] is [tex]\(28^\circ\)[/tex], the "opposite" side is the height of the vertical beam ([tex]\(1.6\)[/tex] meters), and the "adjacent" side is the horizontal distance we need to find. Let's denote the horizontal distance as [tex]\(d\)[/tex].
[tex]\[
\tan(28^\circ) = \frac{{1.6}}{{d}}
\][/tex]
Solving for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{{1.6}}{{\tan(28^\circ)}}
\][/tex]
Using a calculator, we find that:
[tex]\[
\tan(28^\circ) \approx 0.5317
\][/tex]
Therefore,
[tex]\[
d = \frac{{1.6}}{{0.5317}} \approx 3.0091623445541313
\][/tex]
The approximate value of [tex]\(d\)[/tex] is [tex]\(3.0\)[/tex] meters when rounded to one decimal place. Therefore, the correct answer is:
[tex]\[
\boxed{3.0 \text{ meters}}
\][/tex]
