Answer:
The distance between the planes is decreasing at a rate of 614 knots.
Step-by-step explanation:
Let x be the distance A is from the intersection point, and let y be the distance B is from the intersection point. Let s be the distance between A and B, so s² = x² + y²
Note that s, x and y are functions of time, t, so to emphasize this we should write
s(t)² = x(t)² + y(t)²
Differentiate both sides of the above equation, with respect to t, to get
2s*ds/dt = 2x*dx/dt + 2y*dy/dt .
Dividing by 2 gives
s*ds/dt = x*dx/dt + y*dy/dt ,
and dividing by s gives
ds/dt = (1/s)*(x*dx/dt + y*dy/dt)
Since
dx/dt = −442,
dy/dt = −481,
x = 5 nautical miles, and
y = 12 nautical miles, we have
ds/dt = (1/√(5² + 12²))*(5*(−442) + 12*(−481)) = −7982/13 = − 614.
Thus the distance between the planes is decreasing at a rate of 614 knots.