Related Rate Problem (Involving Trig.)

[NATURE.M]

Homework Statement

A rocket is moving into the air with a height function given by h(t) = 200t^2. A camera located 150 m away from the launch site is filming the launch. How fast must the angle of the camera be changing with respect to the horizontal 4 seconds after liftoff?

Homework Equations

The Attempt at a Solution

If we create a diagram, we will see that
tan(θ)=(200t^2)/150 or (4t^2)/3

Differentiating with respect to t,

sec^2(θ)dθ/dt=8t/3 which becomes dθ/dt=8t/3 * cos^2(θ)

At t=4s, tan(θ)=64/3, and then by sinθ=cosθtanθ, we know sinθ=(64/3)cosθ

Then by sin^2(θ)+cos^2(θ)=1, we know that cos^2(θ)=9/4105

Now evaluating the derivative at t=4s, we obtain dθ/dt=96/4105 rad/s≈0.0234 rad/s

I would just like to know if all my steps are accurate, and if my final answer is correct, or if I made an error along the way, leading to an incorrect result?

 

[LCKurtz]

Looks OK to me. You could have saved a couple of calculations by using $\sec^2\theta=1+\tan^2\theta$ instead of messing with the sines and cosines.

 

[NATURE.M]

Okay thanks, and I completely forgot about that identity when doing this problem.