A recording camera is located 2500 feet from the launch pad for recording a rocket launch. As the rocket lifts off, the angle of elevation of the camera increases 4 degrees per second while recording the launch. Find the instantaneous velocity and acceleration of the rocket at time t>0.

using ' to denote d/dt, the height h is

h = 2500 tanθ
h' = 2500 sec^2 θ θ'
at t=0, θ'=4*pi/180
h' = 2500 * 4pi/180 = 5.55 ft/s

h'' = 2500 sec^2θ (2tanθ θ' + θ'')

Thanks, but how do you find the instantaneous velocity from that? I know that acceleration is the derivative from the equation of Instantaneous velocity.. but I don't know how to get the equation for it.

Nevermind, I see what you did. Thank you!

To find the instantaneous velocity and acceleration of the rocket, we need to use the angle of elevation and the rate of change of the angle.

Let's assume that the height of the rocket at time t is h(t) where t > 0. The angle of elevation, θ, can be related to the height and the distance from the camera using trigonometry.

tan(θ) = h(t) / 2500

We want to find the instantaneous velocity and acceleration, so we need to differentiate the above equation with respect to time.

Differentiating both sides of the equation with respect to t:

sec^2(θ) * dθ/dt = dh/dt / 2500

Now, we are given that the angle of elevation increases at a rate of 4 degrees per second:

dθ/dt = 4 degrees/sec = 4 * π/180 radians/sec

Substituting this value into the equation and rearranging:

sec^2(θ) * (4 * π/180) = dh/dt / 2500

Simplifying further:

sec^2(θ) = (dh/dt) * (180/(4 * π)) * 2500

Now, we have an equation that relates the derivative of the height with respect to time (dh/dt) to the angle of elevation and its rate of change. To find the instantaneous velocity and acceleration, we need to evaluate this equation at a specific time t and use the angle of elevation at that time.

Note: The sec^2(θ) term represents the square of the secant function, which can be calculated using the trigonometric identity:

sec^2(θ) = 1 + tan^2(θ)

By substituting this identity into the equation, we can evaluate the expression for sec^2(θ).