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# Analyzing related rates problems: equations (Pythagoras)

Mastering related rates problems hinges on the careful selection of an equation that accurately links the quantities. We'll illustrate this with a diagram, emphasizing the role of the right equation in solving these problems and understanding the interplay of quantities over time.

## Want to join the conversation?

• What does the answer end up being?
• I used two methods to calculate it and got -102.307 (approximated) twice
• why is this so hard
• If you do not feel ready for this, I encourage you to watch the previous videos in the Calculus 1 playlist.
If you still face difficulty, go to the Precalculus playlist.
If you still don't feel ready, go to the Get ready for precalculus playlist.
I hope this helps.
• Why is distance of car to intersection relevant? can this be solved if we just put respective car speeds into pythagorean theorem (as they denote rate of change of car distance to the intersection at point t0). If you do that you get similar result ~102.9km/h
• To get the answer you have to find the instantaneous rate of change of function d(t) at instant t0. To get this value, you would find what the function of d(t) is, get it's derivative, then plug in the values to get your answer. To do this you need the values, d, x(t), and y(t). X(t) and Y(t) are the distances to the intersection, while d can be found using the pythagorean theorem. As you found out, doing it the way you described does not give you an exact answer and if you put that on a test you would most likely get the question wrong.
• Is t sub zero just the time at zero seconds?
• I was wondering if you could set it up as d(t) = √(x(t)^2 +y(t)^2) and then take the derivative. d'(t) = 1/2(x(t)^2+y(t)^2)^-1/2 (2x'(t)+2y'(t)
and then plug in all of your info. Does that work?
• I had the same idea.

I think you are almost correct. I think it should be:
d'(t) = (1/2(x(t)^2+y(t)^2)^(-1/2))(2xx'(t)+2yy'(t))

that is, you forgot to apply the chain rule to (x(t))^2 and (y(t))^2
(1 vote)
• Is this like some sort of function that is defined by its derivative?