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 4000ft 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

To find the rate at which the camera's angle changes during a rocket's ascent, we use related rates in calculus, considering the tangent of the angle, differentiating with respect to time, and solve for the rate of change of the angle in radians per second.

Step-by-step explanation:

To determine at what rate the angle of the camera is changing when the rocket is 4000ft up and rising at 100 feet per second, we can use related rates, which are part of differential calculus.

We assume the angle between the horizontal line from the camera to the rocket, denoted by θ, changes over time. We're given that the horizontal distance (adjacent side) is 3000ft and the vertical distance (opposite side) is increasing at a rate of 100ft/s when the opposite side is 4000ft.

Step-by-step explanation:

Using the tangent function, which relates the opposite side to the adjacent side, we have Τan(θ) = opposite / adjacent.

The rate of change of the angle can be found by differentiating both sides with respect to time (t), where Τan(θ) becomes sec2(θ) • dθ/dt.

Plug in the values for the rate of change of the opposite side (d(opposite)/dt = 100ft/s), the constant adjacent side (3000ft), and find the value of sec(θ) using the current opposite side (4000ft).

Solve for dθ/dt, which is the rate of change of the angle in radians per second.

With the values plugged in, you would find the rate at which the angle is changing and round your answer to the nearest hundredth as required.