Thread: Related Rates & Angle of Elevation Help??

I'm not sure how to do these two questions. Any help with both or one is appreciated!!

1) A balloon rises into the air started at point P. AN observer, 100m from P looks at the balloon and the angle theta between her line of sight and the ground increases at a rate of 1/20 rads per second. Find the velocity of the balloon where theta = pi/4

The answer is 10m/s I'm just not sure how to get it.

2) One end of a ladder of length 5m slides down a verticle wall. When the upper end of the ladder is 3m from the ground, it has a downward velocity of 0.5m/s. Find the rate at which the angle of elevation is changing at that time.

Again, the answer is 0.125 rad/s

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

(1)

We also know by Pythagoras that:

(2)

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.