(Airplane & Radar) An airplane is flying (horizontally) at the height of 6 km on a flight path that will take it
directly over a radar tracking station (on ground). If the distance D between the plane and the radar is
decreasing at a rate of 300 km/hr, find the speed of the plane when D remains 10 km.
x^2 + h^2 = D^2
solve for x when D = 10 and h = 6
x = 8 (3,4,5 triangle)
2 x dx/dt + 2 h dh/dt = 2 D dD/dt
but dh/dt = 0
so
2 (8)v + 0 = 2 (10) 300
v = 3000/8
To find the speed of the plane when the distance D remains 10 km, we need to analyze the relationship between the speed of the plane and the rate at which the distance D changes.
Let's denote the speed of the plane as v (in km/hr). Since the plane is flying horizontally, the rate at which the horizontal distance between the plane and the radar changes is equal to the speed of the plane. Therefore, the rate at which D is decreasing is v (in km/hr).
We are given that the rate at which D is decreasing is 300 km/hr, so we can set up the following equation:
v = -300
The negative sign indicates that the distance D is decreasing. Next, we need to find the relationship between D, the height of the plane, and the horizontal distance between the radar and the plane.
Using Pythagoras' theorem, we can relate the height of the plane (h) and the horizontal distance (x):
D^2 = x^2 + h^2
Differentiating both sides of the equation with respect to time will allow us to find the rates of change:
2D(dD/dt) = 2x(dx/dt) + 2h(dh/dt)
Since the plane is flying horizontally, dx/dt (the rate at which x changes) is equal to v (the speed of the plane in the horizontal direction). The rate at which h changes (dh/dt) is 0 since the height of the plane is not changing.
Substituting these values, we get:
2D(dD/dt) = 2xv
Simplifying the equation further:
dD/dt = xv/D
We are given that dD/dt = -300 km/hr and that D = 10 km. Substituting these values into the equation, we get:
-300 = 10v/10
Simplifying:
-300 = v
Therefore, the speed of the plane when D remains at 10 km is -300 km/hr. Note that the negative sign indicates that the plane is moving towards the radar station.