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Problem 33 Hard Difficulty

The top of a ladder slides down a vertical wall at a rate of $ 0.15 m/s. $ At the moment when the bottom of the ladder is $ 3 m $ from the wall, it slides away from the wall at a rate of $ 0.2 m/s. $ How long is the ladder?

Answer

5 $\mathrm{m}$

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Video Transcript

So in this problem, we want to know, um that when the bottom of the ladder is 3 m from the wall, we want to know how long the latter is given this information. So we have our ladder and we have our wall. Um, we know that the length of the latter l or right here l is constant. Um, we also know that since the latter is falling, we will have that d y d t is gonna be negative because it's going down like this. So we want to represent the length of the water is elsewhere being equal to X squared plus y squared. And that's just Pythagorean theorem. The idea that the high pop news is equal to the sum of the squares of the X direction and the Y direction. So with that in mind, we assume that the latter is going to be touching both surfaces. The coefficients of two when we take the derivative can just be divided off. So really, all that we're left with is that d l d t is equal to, um is equal. Thio two x dx DT plus two. I do y d t. But once again, as we suggested, we can get rid of those twos. So since, uh, the length is constant, what we really have is going to be zero equals x dx DT times Why or a plus y d Y d t So that's our first equation that we're gonna be using when we solve for why we end up getting that on plugging in the other values that we have, we get the zero is equal to three times point to plus why times negative 0.15 With that, we end up getting that since X is three in this case, that why is going to give us four now, Um, with this in mind, since we have are some of the values that we're looking to find, we know that we now have that three squared plus four squared because that's the X and Y values is going to equal r l squared. And we know that the answer to this because the +345 triangle is five. So the length of our ladder is 5 m