solve-the-given-problems-involving-tangent-and-normal-lines-on-a-particular-drawing-a-pulley-wheel-3

Question

Solve the given problems involving tangent and normal lines.
On a particular drawing, a pulley wheel can be described by the equation $x^{2}+y^{2}=100$ (units in $\mathrm{cm}$ ). The pulley belt is directed along the lines $y=-10$ and $4 y-3 x-50=0$ when first and last making contact with the wheel. What are the first and last points on the wheel where the belt makes contact?

Clarissa Noh · Accepted Answer

Hi, this is Claire. So a section 3.3, number 36 of stories for a calculus book. So here we have our function. Why is equal to X squared minus Mexico, for we are given the courtroom at 10 and were asked to find the tension and normal line equation. So here is the attention and normal line. It&#x27;s a linear function of both. So we&#x27;re gonna find the derivative when, um Xs equal to one. So this is if we get the derby tive of X square minus X to the fourth, we will get the slope. So that&#x27;s gonna be minus or X cubed. Ms. We&#x27;re gonna find this one Xs equal to one in this equals negative too. And that is equal to our slope, which is represented by hey. And so if we plug it into this equation, we get why is equal to negative two acts plus B and using the coordinates 10 we get B as equal 222 and our attention equation is going to be cool to why go to negative to X plus two. Then we&#x27;re gonna box are and sir, And if we&#x27;re gonna find the normal line equation. We know that it&#x27;s going to be the soap is going to be perpendicular. So the slope has to be just huh? That&#x27;s gonna be our A And then our we&#x27;re gonna plug it into our equation more equals X plus be. And then this is, um, our equation for the normal line. And since it passes the 0.10 we&#x27;re gonna plug that in and we got B is equal to negative one house. Our equation for a normal line is why is equal to half of bets minus huh?

Amrita Bhasin · Answer

this question gives us why is sign of Ax Plus three. And they asked us to look at the derivative in order to find equations of the lines that are tin gin and normal. So what we know here is that we know the derivative of Sinus co sign, and the derivative of Constant is simply zero. So we know we have thus and we know our m you consider the street a slope is negative one. Therefore we have why is sign of pie Plus three, which is simply zero plus three, which is obviously just three now substituting into the tangent and that stops shooting into the normal. We have our two lines in the

Regina Hays · Answer

Okay. What we want to do is we want to find equations for, um a tangent lines and be, um, normal lines to the or to a or toothy hyperbole. And the hyper Bella is X squared over nine minus. Why squared is equal to one and we want the&#x27;s done at X equal to six. Um, and so tangent lines. Anytime we see Tanja lines where you have to think of, we&#x27;ve think of derivative and the word normal line normal lines. And I forgot the word normal lines up here, um, means that tangent lines and normal lines are perpendicular to each other. Um and then So once I confined this slope of tangent lines, then the slope of the normal lines we&#x27;re just gonna be the negative. Reciprocal. Lt&#x27;s aan dso We&#x27;ve got to think about the derivative. So we&#x27;re gonna take the derivative with respect to X of the hyperbole equation. And so this becomes two X over nine minus two. Why times t y over dx equal to zero. And so if we saw for D y over dx, we get to why over d x is equal to, um X over nine. Why so um, the derivative is dependent on the X value and the Y value and all I&#x27;m given. It&#x27;s the X value. So what I need to also do is to find the corresponding why value for the X equal to six. So four X equal to six. We get 36/9 minus y squared equal toe one which gives me, um why equal the plus and minus the square root of three. And so we&#x27;re gonna have to do this multiple times because we get to different lie values when X is equal to six. And so let&#x27;s go ahead and get started. So, um, then let&#x27;s go ahead. Also in here. Let&#x27;s go ahead. And if, um, let&#x27;s evaluate d y over d eggs at, um X equal to six, and why equal two plus and minus this grow to three. So when I do that, I get plus and minus to Route 3/9. Um, and I just put a six in here in this crew to three and kind of rationalized it. And so the positive, um, derivative goes with the positive. Why value the negative derivative goes with the negative, and so then what we can do is at, um, the 0.6 comma. Positives grow to three. Um, the equation for the line is why minus the square root of three is equal to the positive square to two, um, positives to Times Square to three over over nine. Because that is a slope times X minus six. And so now what we can do is go ahead and solve for why? And so we get Y is equal to two times. Just grow to three over nine times eggs minus the square to 3/3. Or if you want to ride it inthe e um, standard form. This is the why Intercept form. If you want to offend, write it in the standard form, you can get two X minus three times a squirter. Three. Why minus three equal to zero. Um, and all idea was distribute in and simplify. Now, at the point, Sit common six. Common. Negative. Screw it. At three. We get why. Plus, the square root of three is equal to negative to victory over nine times X money. Six. Because don&#x27;t forget. When we, um substituted in the plus square to three, we would get the plus, we would get the plus, um, derivative and the positive derivative. When we evaluated at the negatives grow to three, we get the negative derivative, and that&#x27;s where I&#x27;m getting both of those values I&#x27;m. And then, of course, we can distribute in sulfur. Why? And we get why is equal to, um, negative two times a screw to 3/9 X plus 10 times the square root of three him that that is not what I wanted to do. Um, we we had a positive scurvy to three over three. And then, of course, this is a white intercept. We can change it to that general form, which would give me two x plus this grow tits three times the square to three. Why minus three, equal to zero. Either way, we&#x27;ll do So this is, um, the tangent line. So these are the tangent lines to the curve at X equal to six. And so now what we want to do is to find the normal lines, and so don&#x27;t forget normal and tangent lines are perpendicular to each other. Um, and so the slopes for the normal lines will now be equal to plus and minus three, Route three over to. So I just basically took, um, the slope of the tangent lines and change the signs and flip them. And so and we&#x27;re still looking at the same points. So for, um, the 0.6 square root of three comma square to three we get why minus the square root of three is equal to Now here&#x27;s the deal is when I have a positive, um, slope for the tangent line. My normal line&#x27;s gonna be the negative value. And so we can&#x27;t forget that Times X money six. And of course, we can go ahead and sell for Why? Why is equal the native three. Route 3/2 X Plus 10 Route three. Or we can change it into the general form, which we get nine X plus 2 to 3. Why minus 60 equal to zero. Okay, now for the 0.6 common negative screwed three we have Why. Plus the square D three is equal, and it&#x27;s gonna be the positive this time, because, remember, slopes perpendicular lines have opposite signs. Times X minus six Saw for why we get wise equaled a positive three scrolls or three over to X A minus 10 Route three this time, or you can change it to that general form, which is nine X minus two. Route three. Why minus 60 equal to zero. And so these are the equations for the normal lines at X equal 26

Matthew Lee · Answer

so Ed for finding any tension on in your Know mine need to you smoke on group and it went on this head. So let&#x27;s first find a point on the tide, bro. Come on. Which correspond to where actually going for Are we looking for here? If y squared over four minus 16 over to their very quickly, that&#x27;s going to be eight equals one. The fourth worked exchange about Now move it over the other side much bye bye floor. You move it over, you get nine, then kind of forget you 36. And if meeting two different ones are you question minus six. So they&#x27;re after two points, or X If you go to four, we have four common six and four common negative six. That makes sense because it&#x27;s a curricula. If you were to graph it, you will find that it repeats. It&#x27;s not a function, so they&#x27;re two different pointed. Have a next year, for we&#x27;re happy to this point. And each point there&#x27;s going to be a stroke. So to find smoke, of course, we take the derivative. This equation is not functioning. We&#x27;re going to need it in quickly differentiation. Here So when we take that, I probably question what&#x27;s going on before my next word. Oh, to be a good one. I mean, different shit inside. We&#x27;re going to get to Why, um Why crime When you put General Choi wide crime over for mine it to Iraq over to equal of Europe and simplify did too. Why? Why crime over too. My X equals zero now to find white crunch on the two different points. Then what&#x27;s first plug in the points? Find a smoke at four. Comma six. That what you&#x27;re looking for in six. So we&#x27;re going to get but she fucking six here. What about two? Get three white crime minus four actually going, Vera. And if you stop the white crime and you get for third, this is the slope for this point for comics. Let&#x27;s go ahead and find any, uh, read the tangent line at four. Common six. That&#x27;s for common. Six. The smoke is, uh, m alright. It&#x27;s for third. So, using the pointcorp equation, add with my mom, tenant rights and cockle. If you just simply do why minus six, you called for third documents for it from here. It&#x27;s just invaluable. So he distribute before thirds. Over here, you get 4/3. It&#x27;s your turn. Um, my 16 years And then if you have picked each side, you get for third ex. That&#x27;s easy. Affect your fixes in 18 3rd so adding aging. Third, give that to third. Here the can shine at four comma fix now on a normal line at four comma six for the normal line you the same point. But we&#x27;re gonna use the smoke which perpendicular to this line. And so we&#x27;re going to use the negative reciprocal slope, which is negative reports. They&#x27;re using this folk now, but the same point we&#x27;ll get us in normal line. At this point, we&#x27;re gonna have one minute, six again, have a smoke with negative reports we&#x27;ve been exploited for. If you&#x27;re getting a get reports ex, um, my mystery blustery. And if you have to get outside and optionally to get the equation negative for X last night, So here is the tangent line at four comma six here, the normal line for comics Now do the same thing. But for the point for negative six over this, we&#x27;re gonna need to buy a new smoke you took it back into the quantity equation for the derivative which we had back here. They were looking for common negative six here, then we&#x27;re going to get, um Let&#x27;s see. Yeah, the bug negative. Fixed here. Negative. 12. Sometimes you need to simplify them quick in here. That part of the ethics here we get negative three negative three. Being like crime, I&#x27;m actually still four. So basically the negative sign here instead of a positive so negatively like crime minus 40 it could be very similar. Except that smoke will be negative now, the airport in that. So I divided binding. Agree what you get, uh, negative orders by private. Anything for their 12 at this point. Uh, so again, we can apply the points of the equation very similar to what we just did. Now, the point of a little different worldwide Quest IX. You call it negative 4/3. I&#x27;m excited for We&#x27;re gonna start attention. When were you doing this work here? Uh, the tribute. You get negative 4/3 look, but 16 dirt. Then if you now, we&#x27;re gonna track six from that side of the equation. So from each side, other the negative part of X 68 endurance jackets endured from 16 3rd Give negative. Hubert, You&#x27;re the changeling at this point for the normal line while just take the strobe and put it around. Take the negative. Are typical normal. Perpendicular. Where were you? 34 now. And you can see it&#x27;s gonna be very similar to what we just did. Uh, anything like fix You go on the report. Times X minus four. I&#x27;m going to get one more page. If we distribute that, we get wide bucks. Six. It&#x27;s for you. You see that? You get 34 x my history and then finally tracked chicks. Should we get 34 x mine after nine. To hear the normal line for Grima at this point for common negative. Fix your the tangent line. Uh, hear the tension around here or that problem at this point. So there were two tension lines, or rather, they&#x27;re two different points were active for and each of the points have attention Aligned animal on

Clarissa Noh · Answer

it&#x27;s Clarissa when you right here. So we have acts is equal to two x e to the X To find the derivative, we got D over DX. To exceeds the X, we used the product Rule two x d to the axe eat the X plus eat the X d over DX for two X this becomes equal to eat. The axe comes to X plus two and the slope of detention at 00 becomes too. So the equation of attention is why equals two X? We have to note that while we multiply perpendicular lines, there slope is negative one. So the normal normal line has a slope of negative 1/2. So the equation of the normal line is gonna be Y minus zero over X minus zero, which is equal to negative one tough ex. So this becomes a Y is equal to negative 1/2 X for the normal line.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Kayleah T.

Caleb E.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus