You're on the right track.
A hot air balloon is rising straight up from a level field and is tracked by a range finder 500ft from the lift point. At the moment the range finder's elevation angle is pi/4, the angle is increasing by .14 radians per minute. How fast is the balloon rising at that instant.
So using the the trigonometric laws I know that the current height is equal to tan θ = op/adj which for this problem would be tan pi/4 = opp/500. I can use that to get the height, but can I also use that for the derivative to calculate the height?
I guess I'm not really sure how to relate the change in angle towards the change in height necessarily.
I'm also a little thrown off by the use of radians. Would it be advantageous to convert to degrees and then convert back to radians at the end?
So far the only guess I have at a derivative would be sec^2 θ * dθ/dt = dh/dt / 500, but I'm pretty sure that's pretty wrong.
Why do you doubt yourself?
As you observed, we haveCode:. /| / | / | / h / | / | / θ | --o-------500----+
So, we differentiate implicitly with respect to time ,
When the angle is , we have , so we substitute:
If you carry through the units, you should see that we have feet per minute, so the balloon is rising at a rate of .