To solve this problem, we need to calculate the horizontal distance from the vertical beam to where the lower end of the support beam should be placed. Here's a step-by-step solution:
1. Understand the Problem:
- We have a support beam placed at a [tex]\(28^\circ\)[/tex] angle of elevation.
- The top of this beam meets a vertical beam 1.6 meters above the horizontal floor.
- We need to find the horizontal distance from the vertical beam to the lower end of the support beam.
2. Visualize the Scenario:
- Imagine a right-angled triangle where:
- The vertical beam represents the opposite side (1.6 meters).
- The support beam represents the hypotenuse.
- The horizontal floor distance we need to find represents the adjacent side.
- Given the angle of elevation ([tex]\(28^\circ\)[/tex]), we can use trigonometric ratios to find the horizontal distance.
3. Trigonometric Relationship:
- We use the tangent function, which relates the opposite side and the adjacent side of a right-angled triangle:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, [tex]\(\theta = 28^\circ\)[/tex] and the opposite side is 1.6 meters. We solve for the adjacent side (horizontal distance).
4. Calculate the Horizontal Distance:
- Rearrange the tangent function to solve for the adjacent side (horizontal distance):
[tex]\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}
\][/tex]
[tex]\[
\text{adjacent} = \frac{1.6}{\tan(28^\circ)}
\][/tex]
- Calculating [tex]\(\tan(28^\circ)\)[/tex] in radians:
[tex]\[
\theta \text{ in radians} = 28^\circ \times \frac{\pi}{180} \approx 0.489 \text{ radians}
\][/tex]
- From the previously obtained true result, we find:
[tex]\[
\tan(0.489) \approx 0.5317
\][/tex]
- Now substitute:
[tex]\[
\text{adjacent} = \frac{1.6}{0.5317} \approx 3.009 \text{ meters}
\][/tex]
5. Match with Given Choices:
- The closest value to [tex]\(3.009\)[/tex] meters within the provided choices (3.0 meters, 3.4 meters, 3.9 meters, 4.4 meters) is:
[tex]\[
\boxed{3.0 \text{ meters}}
\][/tex]
Thus, the lower end of the support beam should be placed approximately 3.0 meters away from the vertical beam along the horizontal floor.
