Question

you are stationary on the ground and are watching a bird fly horizontally towards you at a rate of 5 m/s. the bird is located 40 m above your head. how fast does the angle of elevation of your head change when the horizontal distance between you and the bird is 170 m? (leave your answer as an exact number.)

Answer 1

The rate at which the angle of elevation of your head changes when the horizontal distance between you and the bird is 170 m is approximately -0.05914 radians per second.

Step-by-step explanation:

To find the rate at which the angle of elevation of your head changes, we can use trigonometry.

The horizontal distance between you and the bird is 170 m.

The vertical distance between you and the bird is 40 m.

Using the tangent function, we can find the angle of elevation of your head: tan(θ) = vertical distance / horizontal distance.

θ = arctan(40/170)

θ ≈ 13.40°

Next, we differentiate the angle of elevation with respect to time using the chain rule:

d(θ)/dt = d(θ)/dx * dx/dt

Since the bird is flying horizontally, dx/dt = 5 m/s (given).

Next, we find d(θ)/dx and substitute the values:

d(θ)/dx = (1 / (1 + (40/170)^2)) * (0 - 40/170^2) ≈ -0.011828

Finally, we substitute the values into the chain rule equation and calculate d(θ)/dt:

d(θ)/dt = -0.011828 * 5

= -0.05914 radians per second