# Derivative/Rate Question A plane, traveling at a constant height of 5km above the ground, passes directly over a radar station. When the angle from the ground to the plane is (pi/3) radians, the angle is decreasing at a rate of (pi/6) radians/minute. How fast is the plane traveling? The math model that represents this situation is a triangle. The plane has already flown beyond the radar station which is on the ground.  A line segment from the radar station to the plane is the hypotenuse of the triangle.  The vertical distance from the plane to ground is constant...

The math model that represents this situation is a triangle. The plane has already flown beyond the radar station which is on the ground.  A line segment from the radar station to the plane is the hypotenuse of the triangle.  The vertical distance from the plane to ground is constant at 5 km. The horizontal distance (x) changes along the ground.

A relationship between the angle and the sides of the triangle is:

`tan(theta)=y/x`

Since the vertical distance is constant, `tan(theta)=5/x`

Solving for x, `x=5/(tan(theta))`

Take the derivative of both sideswith respect to t:

`dx/dt=(-5*sec^2(theta)*(d theta)/dt)/(tan^2 (theta))`

Substiting in values when theta = pi/3 and d(theta)/dt = -pi/6.

`dx/dt=(-5*sec^2(pi/3)*(-pi/6))/(tan^2(pi/3))`

`dx/dt=(-5*2^2*(-pi/6))/(sqrt(3)^2)`

`dx/dt=((20pi)/6)/3=((10pi)/3)*(1/3)`

`dx/dt=((10pi)/9)=3.491`

The plane is traveling at 10pi/9 or 3.491 km/min.