A support beam needs to be placed at a [tex]$28^{\circ}$[/tex] 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 [tex]$90^{\circ}$[/tex] angle.
Law of sines: [tex]$\frac{\sin (A)}{a}=\frac{\sin (B)}{b}=\frac{\sin (C)}{c}$[/tex]
Approximately how far from the vertical beam should the lower end of the support beam be placed along the horizontal floor?
A. 3.0 meters
B. 3.4 meters
C. 3.9 meters
D. 4.4 meters
Answer (2)
Set up the problem using the tangent function: tan ( 2 8 ∘ ) = d 1.6 .
Rearrange the equation to solve for d : d = t a n ( 2 8 ∘ ) 1.6 .
Calculate the value of d : d ≈ 3.009 .
State the final answer: 3.0 meters.
Explanation
Set up the problem We are given a right triangle formed by the vertical beam, the horizontal floor, and the support beam. The height of the vertical beam is 1.6 meters, and the angle of elevation of the support beam is 2 8 ∘ . We want to find the distance d from the vertical beam to the lower end of the support beam along the horizontal floor. We can use the tangent function to relate the angle of elevation, the height, and the distance: tan ( 2 8 ∘ ) = d 1.6 .
Rearrange the equation To find the distance d , we can rearrange the equation: d = t a n ( 2 8 ∘ ) 1.6 .
Calculate the distance Now, we can calculate the value of d . The result of the operation is approximately 3.009 meters.
State the final answer Therefore, the distance from the vertical beam to the lower end of the support beam along the horizontal floor is approximately 3.0 meters.
Examples
Understanding angles of elevation is crucial in many real-world applications, such as construction and surveying. For instance, when building a ramp, knowing the desired height and angle of elevation allows you to calculate the necessary length of the ramp's base. Similarly, surveyors use angles of elevation to determine the heights of buildings or mountains by measuring the angle to the top from a known distance away. These calculations ensure structures are built safely and accurately.
The distance from the vertical beam to the lower end of the support beam along the horizontal floor is approximately 3.0 meters. This is found using the tangent function relating the height of the beam and the angle of elevation. Thus, the answer is option A.
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