1. ## [SOLVED] Related Rates

i need help with a related rates () problem.

the whole problem is an attachment on this post- it has a diagram and all of the instructions.

i just need help on part c (explanations would be much appreciated), i understand how to do parts a and b

thank you!

2. Dear Reah,
That problem is one of those first problems that you get in the solved examples of rotational kinematics (first course in Mechanics, really). There are two ways to go about it -- one is by analyzing it via calculus and the 2nd is by treating it as a problem in physics. I will answer it in mathematics and tell you where you can refer if you wanna solve it quickly.

Let the angle between the wall and the ladder be $\theta$ and we shall consider the anticlockwise sense as positive, that is, when the angle increases in this sense, the time derivative of the angle, $d\theta/dt$ will be positive and negative otherwise. (In this regard, note the direction of the arrow when I mark $\theta$ in the image I have attached.)

$tan\theta=x/y$
Multiply by y on both sides and differentiate throughout w.r.t. $t$.
So we get:
$dy/dt \tan\theta + y\sec^2\theta\cdot d\theta/dt=dx/dt$
Take $d\theta/dt$ on the left and put all else on the right side of the equality.
Substitute values and get the result. That will be the answer to your third part.

Since you ask for an explanation so I think this should suffice:
As the ladder slips down, the angle $\theta$ clearly increases. Its time derivative is therefore clearly positive. It is this that you have to calculate at a specific instant.
Physics approach!
If you look at it as a purely physics problem you reach the solution pretty quickly.
It will be nice if you study the fifteenth chapter of Vector Mechanics by Beer and Johnston or the equivalent one in Shames' Mechanics, or Miriam and Kraige.
All these are standard mechanics books (available rake-full -- literally -- in college libraries); guess Beer and Johnston is considered the Bible since on of those guys were from MIT. Highly recommend it!

3. Trivua: BTW, the quantity, $d\theta/dt$ is actually the instantaneous (changes every instant!) angular velocity of the slipping rod.