1.) At time , we may state:
Implicitly differentiating with respect to , and solve for
We are given the conditions:
So, use these to find the value of , which represents the vertical speed of the balloon at the given point.
2.) Let x be the distance of the base of the ladder from the wall and y be the height of the top of the ladder on the wall. We may then relate the angle of elevation to these variables with
We also know by Pythagoras that:
Differentiate (1) with respect to and solve for , and replace the resulting trig. function by its ratio definition.
Differentiate (2) with respect to to find .
Now, you should have everything you need to answer the question.