An airplane flies at an altitude of 2 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation θ is changing when the angle is θ = 30°, θ = 60°, and θ = 70°.
I'll do θ = 30°
At that time, the horizontal distance is x, so
cotθ = x/2, so csc^2θ = 1+x^2/4 and x^2 = 12
-csc^2 θ dθ/dt = 1/2 dx/dt
-4 dθ/dt = 300
dθ/dt = -75 rad/hr
To find the rates at which the angle of elevation θ is changing, we need to use trigonometry and differentiation.
Let's start by drawing a diagram to help us visualize the problem. The observer is at point O on the ground, and the airplane is flying at an altitude of 2 miles, which we can represent with a vertical line segment from O to A. The airplane's speed is 600 miles per hour, so we can represent its distance from the observer with a horizontal line segment from A to B.
Now, let's consider the triangle OAB. We are interested in finding the rate at which the angle θ, which is the angle of elevation, is changing. We know that the tangent of θ is equal to the opposite side (OA) divided by the adjacent side (AB).
Using the given information, we have OA = 2 miles, and the distance AB is changing with time. We can use the fact that the speed of the airplane is 600 miles per hour to find the rate at which AB is changing. Let's call this rate dAB/dt.
Now, we can express the tangent of θ as a function of AB:
tan(θ) = OA / AB
Differentiating both sides of this equation with respect to time t, we get:
sec^2(θ) * dθ/dt = -2 / AB^2 * dAB/dt
To solve for the rate at which the angle of elevation θ is changing, we need to find dθ/dt. Rearranging the equation, we have:
dθ/dt = -2 * AB^2 / (OA * sec^2(θ)) * dAB/dt
Now, let's substitute the known values to find the rates at which the angle of elevation θ is changing for θ = 30°, θ = 60°, and θ = 70°.
For θ = 30°:
dθ/dt = -2 * AB^2 / (OA * sec^2(30°)) * dAB/dt
For θ = 60°:
dθ/dt = -2 * AB^2 / (OA * sec^2(60°)) * dAB/dt
For θ = 70°:
dθ/dt = -2 * AB^2 / (OA * sec^2(70°)) * dAB/dt
Note that to find the rate dAB/dt, you would need additional information about how AB is changing with time.