An airplane is flying above an observer standing on the ground. At the moment it passes overhead, the observer judges by its apparent size that the airplane’s altitude is 30,000 feet. It takes 50 seconds for the plane, moving in a straight, horizontal line, to move through an angle of 0.94 radians as seen by the observer. The angle described is swept out by the line segment joining the observer to the plane. At this moment, the angle is increasing at a rate of 34.3 radians per hour. What is the speed of the plane at this time, in miles per hour?

Question

An airplane is flying above an observer standing on the ground. At the moment it passes overhead, the observer judges by its apparent size that the airplane’s altitude is 30,000 feet. It takes 50 seconds for the plane, moving in a straight, horizontal line, to move through an angle of 0.94 radians as seen by the observer. The angle described is swept out by the line segment joining the observer to the plane. At this moment, the angle is increasing at a rate of 34.3 radians per hour. What is the speed of the plane at this time, in miles per hour?

What equation is this looking for?

Answer 1

Im sorry this is calc

Answer 2

As usual, draw a diagram. The horizontal distance x (ft) flown is found using

x/30000 = tan .94
x = 41077 ft

The speed must be changing, or it would be a constant 41077ft/50s = 560 mi/hr. So, since

tan(z) = x/30000
sec^2(z) dz/dt = 1/30000 dx/dt
To get the units right, 30000ft=5.68mi

2.87 * 34.3 = 1/5.68 dx/dt
dx/dt = 559 mi/hr

Huh! All that calculus was for nothing. We found the speed just by figuring the subtended angle and the time. They should not have included that bit about using 50 seconds.

Answer 3

The equation that is being sought in this problem is the relationship between the speed of the plane and the rate at which the angle is changing.

To find this equation, we can use the concept of angular velocity. Angular velocity is the rate at which an object rotates or sweeps out an angle per unit of time. It is usually represented by the Greek letter omega (ω). The relationship between angular velocity, linear velocity, and radius is given by the equation:

v = ωr,

where v is the linear velocity, ω is the angular velocity, and r is the radius of the circular path.

In this problem, the plane is moving in a straight, horizontal line, so there is no circular path. However, the angle described by the line segment joining the observer to the plane is increasing at a certain rate. This can be treated as an angular velocity.

The problem states that the angle is increasing at a rate of 34.3 radians per hour. Therefore, we can determine the angular velocity (ω) as 34.3 radians per hour.

Now, we need to find the linear velocity (v) of the plane. To do this, we need to know the radius (r) of the circular path. In this case, the radius corresponds to the altitude of the plane, which is given as 30,000 feet.

To convert the altitude from feet to miles (since the speed will be measured in miles per hour), we divide it by the conversion factor of 5,280 feet in a mile:

r = 30,000 feet / 5,280 feet/mile
r ≈ 5.68 miles.

Now, we can substitute the known values into the equation v = ωr:

v ≈ 34.3 radians/hour × 5.68 miles
v ≈ 195.424 miles/hour.

Therefore, the speed of the plane at this time is approximately 195.424 miles per hour.