A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 3 mi apart, to be ∠x = 35° and ∠y = 45°, as shown in the figure. (Round your answers to two decimal places.)
review your basic trig functions, and you will that the height h (in miles) can be found using
h cotx - h coty = 3
To answer this question, we need to use the concept of trigonometry.
Let's first understand the situation. A pilot is flying over a straight highway, and he measures the angles of depression to two mileposts. The angles of depression are the angles formed by a horizontal line and the line of sight from the pilot to the mileposts. In this case, the angles are ∠x = 35° and ∠y = 45°.
Now, we need to use these angles to find out some information about the situation. Since we have two angles and a distance between the mileposts, we can use trigonometric ratios to determine the height of the plane.
The tangent ratio is commonly used in situations involving angles of depression. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side represents the height of the plane, and the adjacent side represents the distance between the mileposts.
Let's start with angle ∠x = 35°. The tangent of this angle can be calculated as follows:
tan(35°) = height of plane / distance between mileposts
Similarly, for angle ∠y = 45°, we can use the tangent ratio:
tan(45°) = height of plane / distance between mileposts
To find the height of the plane, we can rearrange these equations. Let's solve for the height of the plane for each angle:
height of plane = distance between mileposts * tan(35°)
height of plane = distance between mileposts * tan(45°)
Now, let's substitute the given values. The distance between the mileposts is 3 miles.
height of plane = 3 * tan(35°)
height of plane = 3 * tan(45°)
To find the answers, we can use a scientific calculator or an online trigonometric calculator to evaluate the tangent of the angles. Once we have the values, we can calculate the heights.
By substituting the values and performing the calculations, we find:
height of plane ≈ 1.99 miles (to two decimal places) for angle ∠x = 35°
height of plane ≈ 3 miles (to two decimal places) for angle ∠y = 45°
So, the height of the plane above the highway is approximately 1.99 miles when the angle of depression is 35° and approximately 3 miles when the angle of depression is 45°.