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**differentiate** a hot air balloon rises straight up from a level field and is detected by a range finder 200 m from the lift off point. At that point, the angle of elevation of the balloon is 60 degrees, and the angle is increasing at the rate of 0.1 rad/min. Find the rate at which the balloon is rising when the angle of elevation of the balloon from the range finder is 45 degrees.

Could someone please show me the full steps

thanks in advance

Code:

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200

$\displaystyle \tan{\theta} = \frac{h}{200}$

$\displaystyle \sec^2{\theta} \cdot \frac{d\theta}{dt} = \frac{1}{200} \cdot \frac{dh}{dt}$

45 degrees is pi/4 rad

60 degrees is pi/3 rad

When the angle is 45 degrees:

$\displaystyle 200 \sec^2{\left(\frac{\pi}{4}\right)} \frac{d\theta}{dt}= \frac{dh}{dt}$

$\displaystyle 400\frac{d\theta}{dt}= \frac{dh}{dt}$

Now use the remaining information for when the angle is 60 degrees to finish the problem with simultaneous equations.

Hope that was helpful

Mathemagister