please help with a related-rates problem!!?

Question

At a given moment, a plane passes directly above a radar station at an altitude of 6 miles. The plane's speed is 500mph. (after drawing a diagram) Suppose that the line through the radar station and the plane makes an angle θ with the horizontal. How fast is θ changing 10 min after the plane passes over the radar station?

Answer 1

Take the situation t hours after the plane passes overhead. The plane is 500t miles horizontally and 6 miles vertically from the radar station, so we will have tan θ = 6 / 500t.

Implicit differentiation gives sec^2 θ dθ/dt = -6 / (500t^2)
and since sec^2 θ = tan^2 θ + 1 we get
[(6 / 500t)^2 + 1] dθ/dt = -6/(500t^2)
At 10 min we have t = 1/6, so 6 / 500t = 36/500 = 0.072, so we get
1.072 dθdt = -0.432
and so dθdt = -0.403 rad/hr = -0.384°/min.

Answer 2

Take the situation t hours after the plane passes overhead. The plane is 500t miles horizontally and 6 miles vertically from the radar station, so we will have tan θ = 6 / 500t.

Implicit differentiation gives sec^2 θ dθ/dt = -6 / (500t^2)
and since sec^2 θ = tan^2 θ + 1 we get
[(6 / 500t)^2 + 1] dθ/dt = -6/(500t^2)
At 10 min we have t = 1/6, so 6 / 500t = 36/500 = 0.072, so we get
1.072 dθdt = -0.432
and so dθdt = -0.403 rad/hr = -0.384°/min.
14 minutes ago