Related rates: at what rate the angle between the the ladder and the ground is changing

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A student wrote us with this question:

Hi, thank you for your videos. You made the ladder against the wall
problem easy to remember without complicating the steps!!! In the
problem I was given it also asks to find at what rate the area of the triangle
formed by the ladder, wall, and ground is changing. And at what rate
the angle between the the ladder and the ground is changing.

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Thanks for asking! We’re glad to know the video was helpful, and are happy to extend your understanding to make sense of these two additional questions. Let’s treat them one at a time.

First, if you haven’t already, we strongly recommend practicing the four different approaches you need to know how to use to solve different related rates problems. They’re all listed on our related rates page, and there are practice problems (with complete solutions, of course) for each type. We’ll be making use of two of these types for your problems, starting with the “Geometric Fact” one:

(1) at what rate the area of the triangle formed by the ladder, wall, and ground is changing

Since the problem asks for the rate at which the area is changing, we’re going to use “a Simple Geometric Fact” (as explained on the related rates page). Specifically, the area of a triangle is

A = \frac{1}{2}xy

where we’re using the same notation as in the video: x is the base of the triangle, and y is its height.

If we take the derivative with respect to time of both sides, we have

\frac{dA}{dt} = \frac{1}{2}\left[\dfrac{dx}{dt}y + x \dfrac{dy}{dt} \right]

I assume the problem gives you \dfrac{dx}{dt}, the rate at which the ladder moves away from the wall, and I also assume that you found the rate the ladder slides down the wall, \dfrac{dy}{dt}, earlier when you followed the video. (But if not, we’re happy to help with that! Just let us know.) If you have all of those quantities, you should be set. :slight_smile: (But if you have questions, please let us know where you’re stuck and we’ll do our best to help.)

(2) And at what rate the angle between the the ladder and the ground is changing.

Since this is asking about an angle, the approach we need (again from the Related Rates page) is “Using a Trig Function.” In fact, this very problem is there, the first in that subsection, “Related Rates: Ladder slides, angle changes.” I hope that will be enough to guide you in developing the solution to your problem, but again, if you need more assistance please just ask.

For now, thanks again for your compliment and questions – and please feel free to submit more!

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Thank you for your help. You definitely made it easier to remember for a test, because you make is simply to follow!
Jennifer

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