Question

An airplane is flying at a constant altitude of 20,000ft directly towards an observer stationed on the ground. At 2:00 PM the observed angle of elevation to the plane is 27degrees. One minute later, the angle of elevation is 51 degrees. What is the plane's approximate ground speed in miles per hour?

Answer 1

To find the airplane's ground speed, we calculate the distances using the angles of elevation and the airplane's altitude, and then convert the difference in distances to miles per hour.

Step-by-step explanation:

To calculate the airplane's ground speed in miles per hour, we can use trigonometry to find the distance traveled in one minute and then convert it to miles per hour. At 2:00 PM, the angle of elevation is 27 degrees, and at 2:01 PM, it is 51 degrees. The altitude of the airplane is constant at 20,000 feet.

Let's call the distance between the observer and the point on the ground directly beneath the airplane at 2:00 PM x, and at 2:01 PM y. From the observer's point of view, we have two right-angled triangles. For the first triangle, at 27 degrees we can use the tangent:

tan(27 degrees) = 20,000 / x

For the second triangle, at 51 degrees

tan(51 degrees) = 20,000 / y

To find x and y, we solve these equations:

x = 20,000 / tan(27 degrees)
y = 20,000 / tan(51 degrees)

The distance the airplane traveled between those two points in one minute is x - y. We convert this distance to miles (since 1 foot = 1/5280 miles), and then the speed to miles per hour (since the time span is 1 minute, we multiply by 60 to convert to hours).

Ground speed = ((x - y) in feet * 1/5280) * 60 miles per hour

This will give us the ground speed of the airplane in miles per hour.