The CN Tower is 553 m tall. A person is south of the tower, in a boat on Lake Ontario at a position that makes an angle of elevation of 100 degrees to the top of the tower. A second person is sitting on a park bench, on a bearing of 060 degrees relative to the tower, observing the top of the tower at an angle of elevation of 9 degrees. Determine the displacement between the two people.
To determine the displacement between the two people, we can use trigonometry and the given information.
Let's start by finding the height at which the second person is observing the top of the tower. We can use the angle of elevation of 9 degrees and the height of the CN Tower, which is 553 m.
First, we calculate the height of the tower above the second person's eye level:
Height above eye level = tower height * tan(angle of elevation)
Height above eye level = 553 m * tan(9 degrees)
Height above eye level ≈ 92.53 m
So, the height at which the second person is observing the top of the tower is approximately 92.53 m above their eye level.
Next, let's determine the distance between the two people. We can use the angle of 100 degrees at the first person's position and the bearing of 060 degrees at the second person's position.
To calculate the displacement, we will create a triangle with the two positions and the top of the tower as vertices.
First, we need to determine the distance from the first person's position to the tower. We can use the angle of elevation of 100 degrees and the height of the tower.
Distance to tower = height of tower / tan(angle of elevation)
Distance to tower = 553 m / tan(100 degrees)
Distance to tower ≈ 1.38 m
Now, we can use trigonometry to find the displacement. Let's consider the triangle formed by the two people's positions and the top of the tower.
We have a known angle of 60 degrees between the direction of the second person's position and the tower (bearing 060 degrees) and a known side opposite to this angle, which is the distance to the tower (1.38 m).
Using the sine rule, the formula to find a side of a triangle given an angle and its opposite side is:
Side / sin(angle) = Opposite side / sin(opposite angle)
Let's label the desired side as "displacement" and the opposite angle as 100 degrees.
Therefore,
displacement / sin(60 degrees) = 1.38 m / sin(100 degrees)
Now, solving for displacement:
displacement = (1.38 m * sin(60 degrees)) / sin(100 degrees)
Plugging in the values, we get:
displacement ≈ 1.37 m
So, the displacement between the two people is approximately 1.37 meters.