An observer (O) spots a plane flying at a 55° angle to his horizontal line of sight. If the plane is flying at an altitude of 21,000 ft., what is the distance (x) from the plane (P) to the observer (O)?
I just see a right-angled triangle where
sin55° = 21000/x
x = 21000/sin55 = appr 25,636.3
To find the distance (x) from the plane to the observer, we can use trigonometry. Specifically, we can use the tangent function since we have the angle and the opposite side length (altitude).
1. Draw a diagram: Start by drawing a diagram that represents the situation. Label the observer as "O" and the plane as "P". Draw a horizontal line from the observer to intersect the line of sight. Label the point where it intersects as "A". Then, draw a vertical line from point A to represent the altitude of the plane. Label the altitude as 21,000 ft.
O
|\
| \
x| \ altitude (21,000 ft)
| \
----|----\-----
A P
2. Identify the right triangle: The right triangle is formed by the line connecting the observer to the plane (OP), the horizontal line of sight (OA), and the vertical line representing the altitude (AP).
3. Use the tangent function: The tangent of an angle is equal to the ratio of the opposite side length to the adjacent side length. In this case, the opposite side length is the altitude (AP) and the adjacent side length is the distance (x) from the observer to the plane.
Therefore, we can use the tangent function to write the equation: tan(55°) = AP / x.
4. Solve for x: Rearrange the equation to solve for x. Multiply both sides by x to isolate x on one side of the equation.
x * tan(55°) = AP
x = AP / tan(55°)
5. Calculate x: Plug in the values for AP (altitude) and tan(55°) into the equation and calculate x.
x = 21,000 ft / tan(55°)
Using a calculator, find the value of tan(55°), and divide the altitude by that value to find x.
Once you perform the calculations, you will get your result for the distance (x) from the plane to the observer.
To find the distance (x) from the plane to the observer, we can use trigonometry. We have a right triangle formed by the observer (O), the plane (P), and the line of sight (OP), with the angle between the horizontal line of sight and the line of sight being 55°.
In this triangle, the altitude of the plane (21,000 ft.) is the side opposite to the angle (O), and we want to find the distance (x), which is the side adjacent to the angle (O).
Using the trigonometric function tangent (tan), we have:
tan(angle) = opposite / adjacent
Substituting the known values:
tan(55°) = 21000 / x
To find x, we can rearrange the equation:
x = 21000 / tan(55°)
Using a calculator, we can determine:
x ≈ 21000 / 1.4281
x ≈ 14679.8 ft
So, the distance from the plane to the observer is approximately 14679.8 ft.
Draw a diagram. On the ground, mark the points
Q directly beneath the plane
R directly beneath the line of flight, in the line of sight.
Then
x^2 = OR^2 + RQ^2 + 21000^2
I think something is missing. We only have a vertical distance, and an angle flat on the ground. Without knowing some of the distances from O to the flight line in the air or on the ground, we're stuck.
Or, maybe I have misinterpreted the language. If so, fix th drawing and solve for the distances involved.