A model airplane 50 feet above the ground is flying away from an observer. Find the angle of elevation theta of the plane as a function of the distance from the observer to the plane. What is theta when the plane is 300 feet from the observer?
sin theta = 50/d
if d = 300
sin theta = 50/300 = 1/6
theta = 9.6 degrees
If I drew a triangle the d would be the hypotenuse?
To find the angle of elevation theta as a function of the distance from the observer to the plane, we can use the concept of trigonometry. In this case, we can use the tangent function.
Let's consider a right triangle formed by the observer, the plane, and a perpendicular line from the observer to the ground. The angle of elevation theta is the angle between the line of sight from the observer to the plane and the horizontal line.
In this triangle, the opposite side is the altitude of the plane (50 feet), and the adjacent side is the distance from the observer to the plane, which we'll call "d."
Now, according to the tangent function:
tan(theta) = opposite/adjacent
Substituting the values:
tan(theta) = 50/d
To find theta, we can take the inverse tangent (arctan) of both sides:
theta = arctan(50/d)
Now, to find the value of theta when the plane is 300 feet from the observer, we substitute the distance "d" with 300 feet:
theta = arctan(50/300)
Using a calculator or an online tool, we can determine the value of theta to be approximately 9.46394 degrees.
Therefore, when the plane is 300 feet from the observer, the angle of elevation theta is approximately 9.46394 degrees.