Answer:
0.0031792338 rad/s
Explanation:
[tex]\theta[/tex] = Angle of elevation
y = Height of balloon
Using trigonometry
[tex]tan\theta=y\dfrac{y}{200}\\\Rightarrow y=200tan\theta[/tex]
Differentiating with respect to t we get
[tex]\dfrac{dy}{dt}=\dfrac{d}{dt}200tan\theta\\\Rightarrow \dfrac{dy}{dt}=200sec^2\theta\dfrac{d\theta}{dt}\\\Rightarrow 100=200sec^2\theta\dfrac{d\theta}{dt}\\\Rightarrow \dfrac{d\theta}{dt}=\dfrac{100}{200sec^2\theta}\\\Rightarrow \dfrac{d\theta}{dt}=\dfrac{1}{2}cos^2\theta[/tex]
Now, with the base at 200 ft and height at 2500 ft
The hypotenuse is
[tex]h=\sqrt{200^2+2500^2}\\\Rightarrow h=2507.98\ ft[/tex]
Now y = 2500 ft
[tex]cos\theta=\dfrac{200}{h}\\\Rightarrow cos\theta=\dfrac{200}{2507.98}=0.07974[/tex]
[tex]\dfrac{d\theta}{dt}=\dfrac{1}{2}\times 0.07974^2\\\Rightarrow \dfrac{d\theta}{dt}=0.0031792338\ rad/s[/tex]
The angle is changing at 0.0031792338 rad/s
