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Answer:
The rate at which the distance from the plane to the station is increasing is when it is 5 miles away from the station is 504 mi/h
Step-by-step explanation:
An illustrative diagram for the scenario is shown in the attachment below.
In the diagram, y represent the altitude, z is the horizontal distance from the plane to the station and x is the distance from the plane to the station.
From the Pythagorean theorem, we can write that
x² = y² + z²
Differentiate this with respect to time t
That is,
=
The rate at which the distance from the plane to the station is increasing is
is the rate at which the altitude is increasing, since the altitude is 2, that is constant,
.
is the rate at which the horizontal distance is increasing which is the speed, that is,
and
Now, we will determine z when x = 5.
From x² = y² + z²
5² = 2² + z²
25 = 4 + z²
z² = 25-4
z² = 21
z =√21
Putting all the values into the equation
Hence, the rate at which the distance from the plane to the station is increasing is when it is 5 miles away from the station is 504 mi/h.