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A15 f ladder leans against wall The bottom of the Iadder Is 5 ft from the wall at time t 0 and slides away trom the wall at a rate of 3ft /secFind the velocily of t...

Question

A15 f ladder leans against wall The bottom of the Iadder Is 5 ft from the wall at time t 0 and slides away trom the wall at a rate of 3ft /secFind the velocily of the top of the ladder at timeThe velocity of ladder at [email protected]:

A15 f ladder leans against wall The bottom of the Iadder Is 5 ft from the wall at time t 0 and slides away trom the wall at a rate of 3ft /sec Find the velocily of the top of the ladder at time The velocity of ladder at time @sec:



Answers

In Exercises $11-14,$ refer to a $5 \cdot m$ ladder sliding down a wall. as in Figures 5 and 6. The variable $h$ is the height of the ladder's top at time $t,$ and $x$ is the distance from the wall to the ladder's bottom.
Assume the bottom slides away from the wall at a rate of $0.8 \mathrm{~m} / \mathrm{s}$. Find the velocity of the top of the ladder at $t=2 \mathrm{~s}$ if the bottom is $1.5 \mathrm{~m}$ from the wall at $t=0 \mathrm{~s}$.

No. The problem of 47 which is of from calculus and it belongs to capture application of derivative or application at the rate of change will have one and the same thing. So then it a world and and this is the surface and there is a letter which is 10 ft long, 10 ft long ladder. Okay, So if this is why and this is X, so letter must follow the Pythagorean tuning bicycle student access square. That's why it's quite equal to 10 square, Which means access Choir Plus was quite equal. 200. Okay, now it is given that the foot of the ladder moving away from the wall at 4 ft per second. So the foot, it is obvious that a foot of the ladder is moving this side top of the ladder will slide down slight downward. So it has given that it's a speed that is DX by D. T is 4 ft per second. Okay, when it is 6 ft from the wall, we have to find Okay, so we have to find d y by tt at certain. First we find everybody t Then we'll be yes, taking care of whatever it will required. So for that little differentiate, there's a question with respect to T that is G by t t Access Square plus y squared They called two D by d t 100. So this is do X dx by DT as to why they were by DT equal to zero. If you divide both sides by two, will be getting indeed x x dx by oddity Plus why they were by DT equal to zero. So in this case, we'll be having y dy DT quanto minus x the expected 80 so d y by DT will be equal to minus x dx by DT by why and now we have value of the experiment at four. So this is minus four X by what? But we have to find the value of the ability at when, when the basis based on the latter is 6 ft from the wall. So, like this, this is 6 ft. So we have to find the value of why Because lazier So why square will be 4300 and this intensity 10 squared minus six square that is 64. So I will be equal to eight. So when Mexico. Six were able to eight, so the viability value of the viability develop it even x equal to six or eight will be minus four in two six by eight minus three, so answer will be minus three. Divided by D T minus 3 m purse again or 3 ft per second story feet per second are 3 ft per second down wall in downward direction. Thank you so much.

I'm going to draw a picture of this. We've got a ladder leaning against a wall so its draw the wall and the floor and then I'll make the ladder. It's just gonna be a line but it's in red, that's the latter. Um It's 10 m long. The bottom of the ladder slides away from the wall. So it's going this way at zero point five meters per second. All right. So that's going to be dx over D. T. Where this is X. And this is why, well, we know that X squared plus Y squared equals 10 squared. So now I can take the derivative with respect to time of all of these two. X the X over D. T plus two. Why do you Y over DT? And the derivative of a constant is zero. And that makes sense because the length of the latter is not changing. So D. L over D. T. Would be zero. Now we're told Dx over D. T. And we're also told that in part A X equals four meters. So I'm going to solve this for Dy over D. T. And that's going to give me the UAE over D. T. Just solving the equation for. Dy over DT is going to be negative two X. The X over D. T. Over two Y. Of course the two's cancel out. Mhm. So if I know X. And I know Dx over D. T. Well I also need to know why but using the equation up above and rearranging that for why? Why is the square root of 10 squared? Which is 100 minus X squared? Okay. So I can substitute that into the equation below. Negative Dx over D. T. Over oh I forgot X over square root of 100 minus X squared. Okay, no, I can just put that into a calculator. So let me do that negative dX over DT which is 0.5 negative 0.5 times X which is four divided by where route of 100 minus four squared. And that gives me negative 0.218217 looks like the back of the book is actually looking for an exact answer. So let me redo redo this. Although it's strange that they're looking for an exact answer because they gave us a decimal, which usually means not looking for an exact answer. But nevertheless let's redo this and try to get an exact answer. Mm Okay. Yeah. Alright, there we go. Okay, so it's negative one half times X which is four over the square root, 100 minus four squared, which is 16 negative one half times four is negative two over 100 minus 16 is 84. Okay, but the squared of 84 um, is the same as two times the square root. Now I just have to divide that by four of 21. So it's negative one over the square root of 21. Mhm. And that's in meters per second. So now what if it's eight m away from the wall? Um So the same thing, I'm going to put this in the calculator, but instead of four I just have to put in eight and eight whoops squared. And that gives me negative two thirds meters per second. And these are negative, indicating that it is going downward. Mhm.

All right. Based on the hint we have this information given and we want to find Dy DT. Well, if we take this formula here and use implicit differentiation, we get that. No, sorry. We get to X D X D T plus two. Why? Dy DT equals 400 is just a constant. So the derivative there is zero. We get this formula. So we have everything we need to find Dy DT except for why we have X. And we have dX DT. But we don't know why. We need to know why. Well, we can find why using just this formula and not differentiating what we get Based on the knowledge that x equals 12. What we get is that why squared? It is Equal to 400 minus 12 squared. Which means why is equal to the square root of 400 -144, which equals the square root Of 256. So that's just 16. So now we know why 16 and now we have everything we need. So back to this formula we plug in 12 for X. We plug in five for DX DT. We plug in 16 for why And now we can solve it. Okay So we get 32. Dy uh whoops We get 32. The Y. D. T. Equals -120. So dy DT Equals -120 over 32 Which equals, let's see, this is negative. 30/8 can still be simplified to negative 15 Over four. That's as simple as it can be. And by the way the units here are the units here are feet per second because this is the rate of change of a distance. All distances are given in feet and all times are given in seconds. So this is the desired rate of chain.

Hello, everyone and welcome. So for these related rate problems, it's best to find, um, to create a diagram. So we have a 13 ft ladder over here leaning against the wall and, um, let's see Jack pulling the foot off the ladder away from the wall at a rate of half feet 0.5 ft per second. And how fast the top of the ladder sliding down the wall when the foot of the ladder is 5 ft from the wall. Okay, okay, so let's go ahead and frustrate our relationship. And the relationship we have right now is, um, expert. So let's call this X. Right. Let's call this why the expert plus y squared is equal to Z squared. So, um, we know the squared is these 13. So we have X squared plus y squared physical to 169 and I was gonna differentiate both sides to get two X, um, ex prime plus wise to why? Why Prime physical zero. So what is our ex cameras or why? And what is their excrement? What is our white prime? Well, extreme is what we're trying to find, and, uh, we know that our our why is going to be 5 ft. We know why Prime is half a foot. We also know that when this is five, we know the high part news is going to be 13, and this is gonna be a 5, 12, 13 triangle. So X is going to be 12 experience. So there we go. Um, so we have 24 x prime plus, or is this five is equal to zero. Subtract five and developed by 24 you get ex prime is able to negative 5/24. Um, or you could say that. Ah, the height. Yeah, is decreasing. Bye. Um, 5/24 feet per second. Okay. Thank you for watching. And I hope helped.


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