An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 314 miles per hour. Find d𝜃 dt (in rad/h) when x = 6 and x = 3.

Question

An airplane flies at an altitude of 5 miles toward a point directly over an observer.

theta(x)=cot^-1(x/5)

The speed of the plane is 314 miles per hour. Find d𝜃/dt (in rad/h) when x = 6 and x = 3. (Round your answers to three decimal places.)

x=6___rad/h

x=3___rad/h

Answer 1

We are given the function θ(x) = cot^-1(x/5)

Recall d/dx cot^-1(x) = -1/(1+x²), so by the chain rule (at a speed of dx/dt = 314 mph):

dθ/dt cot^-1(x/5) = -(1/5)/[1+(x/5)²](dx/dt) = -1/[5(1+x²/25)]*314 = -314/(5+x²/5) = θ'(x)

When x=6:

θ'(6) = -314/(5+6²/5) = -314/(5+36/5) = -314/(61/5) = -314*5/61 ≈ -25.738 rad/h

When x=3:

θ'(3) = -314/(5+3²/5) = -314/(5+9/5) = -314/(34/5) = -314*5/34 ≈ -46.176 rad/h

Hope this helped!