a surveyor 1standing on a highway observed the angle of elevation of the airplain as 22.5°. at the same time, a surveyor 2 observed the angle of elevation of the same airplane to be 30°. if the airplane was 850m directly above a point on the highway, find the distance between the twi surveyors
There are two different cases here,
you don't say if they are on the same side of the plane, or opposite sides of the plane.
(eg. both looking east , or one looking east the other looking west)
I have them on opposite sides,
horizontal distance from plane of #1
tan22.5 = 750/x1
x1 = 750/tan22.5
similarly
x2 = 750/tan30
distance between = x1 + x2
if they are on the same side, you would take
x1 - x2
To find the distance between the two surveyors, we can use trigonometry and create a diagram of the situation.
Let's consider the triangle formed by the two surveyors, the airplane, and the point on the highway directly below the airplane.
First, let's label the angles and sides of the triangle:
- Let A be the position of the first surveyor.
- Let B be the position of the second surveyor.
- Let C be the position of the airplane.
- Let D be the point on the highway directly below the airplane.
- Let x be the distance between the two surveyors (AB).
- Let h be the height of the airplane above the point on the highway (CD).
Now, since we have the angles of elevation, we can label the angles in the triangle:
- Angle CAD is 90 degrees since the airplane is directly above the point on the highway.
- Angle CAB is the angle of elevation observed by the first surveyor (22.5°).
- Angle CBD is the angle of elevation observed by the second surveyor (30°).
To find the distance x between the surveyors, we need to use tangent trigonometric function.
In triangle ABC, we can find the length of side AC using the tangent function:
tan(CAB) = h / x
Rearranging the equation, we get:
x = h / tan(CAB)
Now we can substitute the values given in the problem:
h = 850m (given)
CAB = 22.5° (given)
Using a scientific calculator, calculate the tangent of 22.5°:
tan(22.5°) ≈ 0.414
Substituting the values into the equation, we get:
x = 850m / 0.414 ≈ 2055.56m
Therefore, the distance between the two surveyors is approximately 2055.56 meters.