Sam wants to find the height of a window in a nearby building but it is a cloudy day. Sam puts a mirror on the ground between himself and the building. He tilts it so that when he is standing up he sees the reflection of the window. The base of the mirror is 1.22 meters from his feet and 7.32 meters from the base of the building. Sam's eye is 1.82 meters above the ground. How high up on the building is the window? The angle of the mirror is 56 degrees
If the mirror is tilted an angle x, and the angle θ from the normal is formed by the ray from eye to mirror to window, then we have (if the height of the window is h):
1.82/1.22 = tan(π/2-θ-Ø)
h/7.52 = tan(π/2-θ+Ø)
To find the height of the window, we can use some basic trigonometry. Here's how:
1. First, draw a diagram to better understand the situation. Label the following points: Sam's feet (F), Sam's eye level (E), the base of the mirror (B), and the window's height on the building (W). Also, label the distances mentioned in the problem: 1.22m (distance from F to B) and 7.32m (distance from B to W).
2. Now, let's find the distance from Sam's eye level to the window's height. Since the mirror reflects light at the same angle it hits the mirror, we know that the angle of incidence (angle between FE and FB) is equal to the angle of reflection (angle between EB and WB). Therefore, we have an isosceles triangle FEB, where angle FEB is 56 degrees.
3. To calculate the distance EW, we can use the tangent function. The tangent of an angle θ is defined as the opposite side divided by the adjacent side. In our case, the opposite side is ŴB (the height we want to find) and the adjacent side is EB (the distance from Sam's eye to the mirror). So we can write the equation as tan(56 degrees) = ŴB / EB.
4. Rearrange the equation to solve for ŴB: ŴB = tan(56 degrees) * EB.
5. Substitute the given values into the equation: ŴB = tan(56 degrees) * 7.32m.
6. Calculate the result: ŴB ≈ 7.95m.
Therefore, the height of the window on the building is approximately 7.95 meters.