An airplane at a constant altitude of 2 miles flies a horizontal distance toward you at a constant velocity. At the start of your observation, the angle of elevation is 40°. Fifteen seconds later, the angle of elevation is 50°. What is the approximate velocity of the airplane in miles per minute?
Let's assume the velocity of the airplane is v miles per minute.
In the first 15 seconds, the angle of elevation increases from 40° to 50°, so the airplane rises by a height of h.
To find h, we can use the tangent function:
tan(40°) = h/2
h = 2 * tan(40°) ≈ 1.376 miles
In 15 seconds, the airplane travels a horizontal distance of d.
To find d, we can use the cosine function:
cos(40°) = d/2
d = 2 * cos(40°) ≈ 1.532 miles
Therefore, in 15 seconds, the airplane travels a total distance of s = √(d^2 + h^2) miles:
s = √(1.532^2 + 1.376^2) ≈ 2.069 miles
Since the velocity of the airplane is v miles per minute, we can calculate the velocity using the formula v = s / (15/60) = s / 0.25:
v = 2.069 / 0.25 ≈ 8.276 miles per minute
Hence, the approximate velocity of the airplane is 8.276 miles per minute. Answer: \boxed{8.276}.