Answered

A support beam needs to be placed at a 28° 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 90° angle.



Law of sines: $\frac{\sin ( A )}{a}=\frac{\sin ( B )}{b}=\frac{\sin ( C )}{c}$



Approximately how far from the vertical beam should the lower end of the support beam be placed along the horizontal floor?


A3.9 meters
B3.4 meters
C3.0 meters
D4.4 meters

Verified Answer :

  • Identify the right-angled triangle formed by the vertical beam, floor, and support beam.
  • Set up the trigonometric relationship \(\tan(28^\circ) = \frac{1.6}{x}\) using the opposite and adjacent sides.
  • Solve for \(x = \frac{1.6}{\tan(28^\circ)} \approx 3.0 \text{ meters}\).
  • Select option c as the correct distance.
  • The final answer is \(\boxed{3.0 \text{ meters}}\).

Explanation

  1. Problem Analysis

Identify the geometric setup: We have a right-angled triangle formed by a vertical beam (height \(1.6\) meters), the floor, and a support beam. The vertical beam and the floor meet at a \(90^\circ\) angle. The support beam makes a \(28^\circ\) angle of elevation with the floor. We need to find the distance \(x\) along the floor from the base of the vertical beam to where the support beam meets the floor.

  1. Setting up the Equation

In this right triangle, let the angle of elevation be \(\theta = 28^\circ\). The vertical beam is the side opposite to \(\theta\), so the opposite side length is \(O = 1.6\) meters. The horizontal floor distance we want to find is the side adjacent to \(\theta\), so the adjacent side length is \(A = x\). The relationship between these sides and the angle is given by the tangent function:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{O}{A}\]
Substituting our values, we get:
\[\tan(28^\circ) = \frac{1.6}{x}\]

  1. Calculation and Conclusion

Rearranging the equation to solve for \(x\):
\[x = \frac{1.6}{\tan(28^\circ)}\]
Using the calculated value of \(\tan(28^\circ) \approx 0.5317\):
\[x \approx \frac{1.6}{0.5317} \approx 3.009 \text{ meters}\]
Rounding to one decimal place, the distance is approximately \(3.0\) meters. This matches option c.

Examples

In construction or carpentry, understanding these geometric relationships is vital. For example, if you are building a support structure like a rafter for a roof or a brace for a deck, you must calculate the exact placement of your supports to ensure structural integrity. By applying trigonometric ratios, you can accurately determine the horizontal distance required for a given height and angle, ensuring your construction project is both safe and precise.

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