Answer:
0.292 rad/s
Step-by-step explanation:
[tex]b=\text{Distance of rocket from your feet}=20\ \text{ft}[/tex]
[tex]\dfrac{dp}{dt}=\text{Rate of change of height}=19\ \text{ft/s}[/tex]
[tex]p=\text{Distance of the rocket from the ground}=30\ \text{ft}[/tex]
Finding distance between the feet and when the rocket is 30 ft in the air
[tex]d=\sqrt{b^2+p^2}\\\Rightarrow d=\sqrt{20^2+30^2}\\\Rightarrow d=36.1\ \text{ft}[/tex]
[tex]\tan\theta=\dfrac{p}{b}=\dfrac{p}{20}[/tex]
Differentiating with respect to time
[tex]\sec^2\theta\dfrac{d\theta}{dt}=\dfrac{1}{20}\dfrac{dp}{dt}\\\Rightarrow \dfrac{d\theta}{dt}=\dfrac{\cos^2\theta}{20}\dfrac{dp}{dt}[/tex]
[tex]\cos\theta=\dfrac{b}{h}\\\Rightarrow \cos\theta=\dfrac{20}{36.1}\\\Rightarrow \cos^2\theta=\dfrac{20^2}{36.1^2}[/tex]
[tex]\dfrac{d\theta}{dt}=\dfrac{\dfrac{20^2}{36.1^2}}{20}\times 19\\\Rightarrow \dfrac{d\theta}{dt}=0.292\ \text{rad/s}[/tex]
Rate of change of the angle is 0.292 rad/s.
