The angle of elevation from L to K measures 24 degrees. If KL = 19, find JL.
The Correct Answer and Explanation is:
To find ( JL ) in the triangle ( JKL ) where ( L ) is the point of observation, ( K ) is the point above ( L ) at an angle of elevation of ( 24^\circ ), and ( KL = 19 ), we can utilize trigonometric principles.
Step-by-step Solution
- Identify the Right Triangle: The triangle formed by points ( J ), ( K ), and ( L ) is a right triangle. Here, ( KL ) is the vertical side, ( JL ) is the horizontal side, and ( JK ) is the hypotenuse.
- Label the Triangle:
- ( K ): The top point directly above ( L ).
- ( L ): The point of observation at ground level.
- ( J ): The point where the horizontal distance ( JL ) extends.
- Apply the Trigonometric Function: Since we know the angle of elevation ( \angle K ) and the length of ( KL ), we can use the tangent function, which relates the angle to the opposite side (height, ( KL )) and adjacent side (base, ( JL )):
[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{KL}{JL}
] - Substituting Values: Given ( KL = 19 ) and ( \theta = 24^\circ ):
[
\tan(24^\circ) = \frac{19}{JL}
] - Calculating JL: Rearranging the equation to solve for ( JL ):
[
JL = \frac{19}{\tan(24^\circ)}
] - Using a Calculator:
- First, calculate ( \tan(24^\circ) ), which is approximately ( 0.4452 ).
- Substitute this value into the equation:
[
JL = \frac{19}{0.4452} \approx 42.6
]
Conclusion
Therefore, the length of ( JL ) is approximately ( 42.6 ) units. This calculation provides an understanding of how trigonometry can be applied to real-world problems involving angles of elevation and horizontal distances. The tangent function is particularly useful when determining unknown lengths in right triangles where angles and one side are known.
In summary, by using trigonometric principles and the properties of right triangles, we have effectively calculated the distance ( JL ) based on the given angle of elevation and the height above the point of observation.