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?
2 answers
Im sorry this is calc
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.
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.
1 answer