A weather balloonV, rising vertically from a pointPon the ground, is observed from a pointOon the ground, 90mfrom P. Letθbe the angle betweenOPandOV(i.e. the angle between the ground and the sightline to the balloon). At what rate is the balloon rising whenθ=45˚ andθis increasing at 1˚/second?

So, I'm thinking the job is to find:(

dVP)/dtat the time whendθ/dt=1.

We already know that

VP(θ)=90tanθ

and

θ(VP)=arctan(VP/90)

So I take the derivative of both functions, accounting for the fact that bothVPandθare functions of time as well as of each other:

Knowing thatdθ/dt=1 andOP=90, I get that:

Which is 1/180. I notice that if I flip this over and solve ford(VP)/dtin the other equation, I get that

d(VP)/dt=1

Is this right?? If so, should I have been able to see the from the start? A lot of steps in this one, and I feel like I'm kinda guessing as to what I should do.