To determine how far from the vertical beam the lower end of the support beam should be placed along the horizontal floor, let's break down the problem using trigonometry.
We are given the following information:
1. The angle of elevation (A) is [tex]\( 28^\circ \)[/tex].
2. The height of the vertical beam above the horizontal floor (opposite to angle A) is 1.6 meters.
We need to find the distance from the vertical beam to the lower end of the support beam along the floor (let's call this distance [tex]\( d \)[/tex]).
Since this forms a right triangle, we can use the tangent function, which relates the angle of elevation, the opposite side (height of the vertical beam), and the adjacent side (distance along the horizontal floor, [tex]\( d \)[/tex]):
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In our case:
- [tex]\(\theta = 28^\circ\)[/tex]
- Opposite side = height of the vertical beam = 1.6 meters
- Adjacent side = distance [tex]\( d \)[/tex]
Therefore,
[tex]\[
\tan(28^\circ) = \frac{1.6}{d}
\][/tex]
To solve for [tex]\( d \)[/tex], we rearrange the equation:
[tex]\[
d = \frac{1.6}{\tan(28^\circ)}
\][/tex]
Using the provided answer, the tangent of [tex]\( 28^\circ \)[/tex] in radians is approximately 0.4886921905584123 radians (since 28 degrees converted to radians is 0.4886921905584123 radians). We know:
[tex]\[
d \approx \frac{1.6}{\tan(28^\circ)} \approx 3.0091623445541313 \text{ meters}
\][/tex]
So, the approximate distance from the vertical beam to the lower end of the support beam along the horizontal floor should be:
[tex]\[
\boxed{3.0 \text{ meters}}
\][/tex]
Hence, the correct answer is 3.0 meters.
