At a given moment, a plane passes directly above a radar station at an altitude of 7 miles.
(a) If the plane's speed is 400 mph, how fast is the distance between the plane and the station changing half an hour later?
(b) How fast is the distance between the plane and the station changing when the plane passes directly above the station?
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To solve this problem, we need to use the concept of related rates. First, let's define some variables:
Let's call the distance between the plane and the station at any given time "d" (in miles).
Let's call the rate at which the distance between the plane and the station is changing with respect to time "ddt" (the derivative of "d" with respect to "t").
Let's call the time "t" (in hours).
We are given the following information:
The altitude of the plane is 7 miles.
The speed of the plane is 400 mph.
Now, let's solve the problem step by step:
(a) We need to find how fast the distance between the plane and the station is changing half an hour later.
To answer this, we'll differentiate "d" (the distance) with respect to "t" (time):
ddt = dd/dr * dr/dt
The first term on the right-hand side represents the rate at which the distance between the plane and the station is changing with respect to the plane's height, and the second term represents the rate at which the plane's height is changing with respect to time.
Since the plane is flying directly above the station, the distance between them is the hypotenuse of a right triangle. The height of the triangle is 7 miles, and the plane's speed is 400 mph. So, the rate at which the distance between the plane and the station is changing with respect to the plane's height is:
dd/dr = sqrt(1 + (dr/dt)^2)
The plane's speed is given as 400 mph, so:
dr/dt = 400 mph
Substituting these values into the equation, we get:
ddt = sqrt(1 + (400 mph)^2) * (400 mph)
Simplifying, we have:
ddt = sqrt(1 + 160000) * 400 mph
Calculating this expression will give you the rate at which the distance between the plane and the station is changing half an hour later.
(b) We need to find how fast the distance between the plane and the station is changing when the plane passes directly above the station.
When the plane passes directly above the station, the distance between them is the altitude of the plane, which is 7 miles. So, at that moment, the distance is not changing. Therefore, the rate at which the distance between the plane and the station is changing is 0.
In summary:
(a) To calculate how fast the distance between the plane and the station is changing half an hour later, substitute the given values into the formula: ddt = sqrt(1 + (400 mph)^2) * 400 mph.
(b) When the plane passes directly above the station, the rate at which the distance between the plane and the station is changing is 0.