Given:
[tex]\sin\theta=\frac{x}{z}[/tex][tex]\begin{gathered} \frac{dx}{dt}=-60mph \\ \\ z=2miles \\ \\ \theta=\frac{\pi}{6} \\ \\ \frac{dz}{dt}=-55mph \end{gathered}[/tex]
Required:
To find the value of
[tex]\frac{d\theta}{dt}[/tex]
Explanation:
Consider
[tex]\sin\theta=\frac{x}{z}[/tex]
Differentiate with respect to t, we get
[tex]\cos\theta\frac{d\theta}{dt}=\frac{x\frac{dz}{dt}-z\frac{dx}{dt}}{z^2}[/tex]
Now by substituting the values,
[tex]\begin{gathered} \cos\frac{\pi}{6}\frac{d\theta}{dt}=\frac{x(-55)-2(-60)}{2^2} \\ \\ \frac{\sqrt{3}}{2}\frac{d\theta}{dt}=\frac{-55x+120}{4} \\ \\ \frac{d\theta}{dt}=\frac{2}{\sqrt{3}}(\frac{-55x+120}{4}) \\ \\ \frac{d\theta}{dt}=\frac{-55x+120}{2\sqrt{3}} \end{gathered}[/tex]
