an observer 1.5meters tall is 20.5 metres away from a tower 22 metres high. determine the angle of elevation of the top of the tower from the eye of the observer
assuming the eye is on the top of the observer's head, it is clear that
(22-1.5)/20.5 = tanθ
so, what angle θ has tanθ = 1?
To determine the angle of elevation of the top of the tower from the eye of the observer, you can use trigonometry and the concept of similar triangles.
In this scenario, we have a right triangle formed by the height of the tower (22 meters), the distance from the observer to the tower (20.5 meters), and the line of sight from the eye of the observer to the top of the tower.
Now, let's label the sides of the triangle:
- The side opposite to the angle of elevation is the height of the tower (22 meters).
- The side adjacent to the angle of elevation is the distance from the observer to the tower (20.5 meters).
- The side opposite to the right angle is the height of the observer (1.5 meters).
We can use the tangent function to find the angle of elevation (θ). The tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle.
So, tan(θ) = height of the tower / distance from the observer to the tower
tan(θ) = 22 / 20.5
To find the angle θ, we can take the inverse tangent (arctan) of both sides of the equation:
θ = arctan(22 / 20.5)
Using a calculator, you can now determine the angle of elevation (θ) by finding the inverse tangent of the ratio, which gives the angle in radians. To convert it to degrees, multiply by 180/π (approximately 57.3 degrees).
θ ≈ arctan(22 / 20.5) ≈ 46.07 degrees
Therefore, the angle of elevation of the top of the tower from the eye of the observer is approximately 46.07 degrees.