how fast observer's viewing angle is changing

Question

plane is flying at a rate of 200 miles per hr.5 miles directly above the building. If the plane continues to fly fast, how fast observer's viewing angle changing in radians per hour, when the plane is seen at an angle 45 degrees

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Answer 1

Malgorzata, the fisrt step is to draw a picture of the situation with vertical distance being 0.5mi, the plane is at a point in which the distance form the observer to the plane makes an angle of 45 degrees with respect to the vertical distance.

 

Write an equation relating the horizontal distance as a function of the observer's viewing angle in which is:

 

Let x = the horizontal distance in which dx/dt= the planes speed=200 mi/hr

x=0.5*tanθ in which θ=the angle.

Take the derivative of x with respect to time.

 

dx/dt=0.5*sec²(θ)*dθ/dt This implies dθ/dt=1/(0.5*sec²(θ))*dx/dt=2cos²(θ)*dx/dt

 

dθ/dt |_{θ=45 degrees}=2cos²(45)*200=2*1/2*200=200 radians/hr.

 

Hope this helps.