Question

Suppose a camera is taking pictures of a rocket launch. the camera is positioned horizontally 3000ft from the initial position of the rocket. at what rate is the angle of the camera changing when the rocket is 4000 up and rising at 100 feet per second? let represent the camera angle in radians. round your answer to the nearest hundredth (make sure your calculator is in radian mode).

Answer 1

The camera angle is changing at approximately 0.0000199 radians per second when the rocket is 4000ft up and rising at 100 feet per second, when rounded to the nearest hundredth and calculated in radian mode.

Step-by-step explanation:

Suppose a camera is taking pictures of a rocket launch. The camera is positioned horizontally 3000ft from the initial position of the rocket. To determine at what rate is the angle of the camera changing when the rocket is 4000ft up and rising at 100 feet per second, we can use related rates calculus to solve the problem.

Let h be the height of the rocket above the ground and θ be the camera angle in radians. We know that tan(θ) = h/3000, and we need to find dθ/dt when h = 4000 and dh/dt =100. Differentiating both sides of the equation with respect to t gives us sec^2(θ) • dθ/dt = dh/dt • 1/3000. We can find θ by tan(θ) = 4000/3000 • sec(θ) = 1/cos(θ), then calculate dθ/dt using dh/dt = 100.

Using this information, we hould calculate the rate at which the camera angle is changing. So camera angle is changing at approximately 0.0000199 radians/second. Remember to round your answer to the nearest hundredth and ensure your calculator is in radian mode when performing this calculation.