an airplane flies directly overhead at 540mph. 1 minute later is seen at an angle of elevation of 34 degrees. how far did the airplane fly during that time? what is the elevation of the plane?
make a sketch.
let P be the position of the observer and point A the position of the plane directly above him
let B be the position of the plane 1 minute later
let the height AP be h miles
mark the angle of elevation as 34°, then angle PBA = 34°
in 1 minute the plane has traveled (1/60)540 or 9 miles
so AB = 9
h/9 = tan34°
h = 9tan34° = 6.07 miles
height of plane = 6.07 miles
distance covered in 1 minute = 9 miles
To determine how far the airplane flew during that time and the elevation of the plane, we can use trigonometry.
Let's start by calculating the distance the airplane traveled. We know that the speed of the airplane is 540 miles per hour, and 1 minute has passed.
To find the distance traveled, we need to convert the time from minutes to hours. Since there are 60 minutes in an hour, 1 minute is equal to 1/60 hours.
Distance Traveled = Speed × Time
= 540 mph × (1/60) hr
= 9 miles
Therefore, the airplane flew 9 miles during that time.
Now, let's move on to calculating the elevation of the plane. We are given that the angle of elevation is 34 degrees.
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the plane, and the adjacent side is the distance traveled.
tan(angle) = opposite/adjacent
We'll use the tangent function to find the height of the plane.
tan(34 degrees) = height/9 miles
Rearranging the equation:
height = tan(34 degrees) × 9 miles
Using a calculator, we can find the value of tan(34 degrees):
tan(34 degrees) ≈ 0.6682
height ≈ 0.6682 × 9 miles
≈ 6.014 miles
Therefore, the elevation of the plane is approximately 6.014 miles.