# Thread: Hot Air balloon related rates

1. ## Hot Air balloon related rates

problem : A hot air balloon rising straight up froma level field is tracked by a range fider 500 ft from the lift off point. At the moment the range finder's elevation angle is 0.25pi and the angle is increasing at the rate of 0.14 radian per minute. how fast is the balloon rising at that moment?

* It would be really helpful if someone could describe the steps please

2. ## Rates of change

Hello cruxkitty
Originally Posted by cruxkitty
problem : A hot air balloon rising straight up froma level field is tracked by a range fider 500 ft from the lift off point. At the moment the range finder's elevation angle is 0.25pi and the angle is increasing at the rate of 0.14 radian per minute. how fast is the balloon rising at that moment?

* It would be really helpful if someone could describe the steps please
Suppose that at time $t$ minutes, the angle of elevation of the balloon from the range finder is $\theta$, and that the height of the balloon is $h$ feet.

Then $h = 500\tan\theta$ (1)

Now the rate at which the balloon is rising is $\frac{dh}{dt}$ feet per minute.

So, find an expression for $\frac{dh}{dt}$ by differentiating (1) with respect to $t$. (You'll need to differentiate the RHS with respect to $\theta$ first and then multiply the result by $\frac{d\theta}{dt}$.)

Now $\frac{d\theta}{dt}$ represents the rate at which $\theta$ is increasing in radians per minute. And you know the value of this when $\theta = \frac{\pi}{4}$, don't you? So ...?

Can you see what to do now?

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### a hot air balloon rising straight up from a level field is tracked by a range finder 500ft from the lift off point. the angle is increasing at the rate of 0.14 rad per min

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