{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>


a. What is the rate of change of the angle of elevation dθ/dx when the plane is x=500 m past the observer?

Verified step by step guidance

1. Identify the relationship between the angle of elevation θ, the height of the plane (h = 400 m), and the horizontal distance from the observer to the plane (x). Use the tangent function: tan(θ) = h/x.
2. Differentiate both sides of the equation tan(θ) = h/x with respect to time (t) using implicit differentiation. Remember that h is constant, so its derivative is 0.
3. Apply the chain rule to differentiate the left side: d(tan(θ))/dt = sec²(θ) * dθ/dt, and for the right side, use the quotient rule to differentiate h/x.
4. Substitute the known values into the differentiated equation. At x = 500 m, calculate the angle θ using the inverse tangent function: θ = arctan(400/500).
5. Solve for dθ/dt by isolating it in the differentiated equation and substituting the values for x, h, and the speed of the plane (70 m/s) to find the rate of change of the angle of elevation.