The observer and the balloon form a right triangle (see attachment).
The balloon is rising at 1.2 rad per min
Using tangent rule, we have:
[tex]\mathbf{tan(\theta) = \frac{y}{x}}[/tex]
Substitute 3 for x
[tex]\mathbf{tan(\theta) = \frac{y}{3}}[/tex]
Differentiate both sides with respect to time (t)
[tex]\mathbf{sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{3} \cdot \frac{dy}{dt}}[/tex]
Make dy/dt, the subject
[tex]\mathbf{\frac{dy}{dt} = 3 \cdot sec^2(\theta) \cdot \frac{d\theta}{dt} }[/tex]
Substitute [tex]\mathbf{\theta = \frac{\pi}{3}\ and\ \frac{d\theta}{dt} = 0.1}[/tex]
So, we have:
[tex]\mathbf{\frac{dy}{dt} = 3 \cdot sec^2(\pi/3) \cdot 0.1 }[/tex]
Using a calculator, we have:
[tex]\mathbf{\frac{dy}{dt} = 3 \cdot 2^2 \cdot 0.1 }[/tex]
Multiply
[tex]\mathbf{\frac{dy}{dt} = 1.2}[/tex]
Hence, the balloon is rising at 1.2 rad per min
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