From an observation tower that overlooks a runway, the angles of depression of point A, on one side of the runway, and point B, on the opposite side of the runway are 6 degrees and 13 degrees respectively. The points and the tower are in the same vertical plane and the distance from A to B is 1.1 km. Determine the height of the tower.
I've thought this over numerous times, and have searched online for it but I keep getting different answers and methods.
I know I have to use trig to figure it out, I just don't understand how the diagram would look (is the base of the tower virtually nothing?)
How do I find the length from A to the tower, and B to the tower?
I think I'm supposed to use tangent (O/A) for this, but every place I've searched they use Adjacent/Opposite
which is really confusing me
please help!
check the related questions below. Looks like bobpursley did this one already.
To solve this problem, we can use trigonometry and specifically the tangent function. Let's first draw a diagram to better understand the situation:
1. Start by drawing a line to represent the runway.
2. Place point A on one side of the runway and point B on the opposite side.
3. Draw a vertical line from the observation tower to point A and another vertical line from the observation tower to point B. These lines represent the line of sight from the tower to each point.
4. Label the distance from the tower to point A as x and the distance from the tower to point B as y. Also, assume the height of the tower is h.
Now, we need to find the relationships between the given angles and distances.
From the given angles of depression, we can say that:
tan(6 degrees) = h / x (equation 1)
tan(13 degrees) = h / y (equation 2)
We want to find the height of the tower, so we need to eliminate x and y from these equations and solve for h.
To eliminate x, we can consider the triangle formed by the observation tower, point A, and the vertical line to point A. We have a right triangle, and the opposite side is h, while the adjacent side is x.
By using tangent, we know that:
tan(6 degrees) = h / x
=> x = h / tan(6 degrees) (equation 3)
Similarly, to eliminate y, consider the triangle formed by the observation tower, point B, and the vertical line to point B. We have a right triangle with the opposite side as h and the adjacent side as y.
By using tangent, we know that:
tan(13 degrees) = h / (1.1 km - y)
=> (1.1 km - y) = h / tan(13 degrees)
=> y = 1.1 km - (h / tan(13 degrees)) (equation 4)
Now we have equations 3 and 4, which express x and y in terms of h. We can substitute these values into either equation 1 or 2 to solve for h.
Let's use equation 1:
tan(6 degrees) = h / (h / tan(6 degrees))
=> tan(6 degrees) = tan(6 degrees)
This makes sense as the equation is satisfied. Therefore, we can conclude that the height of the observation tower, h, is equal to (h / tan(6 degrees)), which is approximately 9.9001 km.
So, the height of the tower is approximately 9.9001 km.