Final answer:
To find the height for the spotlight installation, the tangent of the angle of elevation (which equals the angle of depression) is used. Multiplying the tangent of 27 degrees by the distance of 11 meters, we find the height to be approximately 5.93 meters, thus the closest answer is 7 meters.
Step-by-step explanation:
To determine the height on the wall where the spotlight should be installed, we can use trigonometric functions to find the height based on the angle of depression and the distance from the wall. Since the angle of depression is 27 degrees and the distance from the spot to the wall is 11 meters, we are dealing with a right-angled triangle where the angle of depression is equal to the angle of elevation from the ground to the spotlight. We want to find the opposite side of the angle, which is the height of the wall where the spotlight will be.
The trigonometric function we will use is the tangent: tan(θ) = opposite / adjacent, where θ is the angle of elevation, the opposite is the height we want to find, and the adjacent is the distance from the spot to the wall.
tan(27 degrees) = height / 11 m
Height = tan(27 degrees) × 11 m
Solving for the height:
Height = tan(27 degrees) × 11 m ≈ 5.93 m
The closest answer choice to 5.93 m is option b. 7 m. Therefore, the spotlight should be installed at 7 meters high on the wall.