let-a-and-b-be-positive-numbers-find-the-length-of-the-shortest-line-segment-that-is-cut-off-by-the-

Name: let a and b be positive numbers find the length of the shortest line segment that is cut off by the
Uploaded: 2019-11-17
Duration: 1 min 23 s
Channel: Numerade
Description: Let $ a $ and $ b $ be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $ (a, b) $.

Question

Let $ a $ and $ b $ be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $ (a, b) $.

Amrita Bhasin · Accepted Answer

Okay. In order to find these shortest length, we know we&#x27;re gonna have to use the formula. Why? I&#x27;m honest. B equals times X minus a, which in terms of acts is a minus. B over. Now recall what the distance formula is. Score this value. Take the derivative. It&#x27;s a bit long, so that&#x27;s equal to zero. And we get, um, is be over. I therefore we know. Do you wet the shortest length of the line segment would be.

James Kiss · Answer

way. Want to find the distance in mid point between the points A, B and B A. Find the distance would be the square roof of the difference of your wife. Ordinate or export its, um a minus B squared. Plus the difference of your, uh why corn? It&#x27;s a B square. The what? It&#x27;s not gonna matter because we spare. It&#x27;s gonna be positive. So we get a swear minus to a B plus B squared, plus a square minus two a B plus B squared. So we get to a square minus for a B plus two he square or the square root of two times a my SP squared, which is the absolutely outing of a minus bi. I&#x27;m suspect root of two because we&#x27;re square rooting this eight square or the midpoint. You just add your exports together a plus B. Take, uh, and you add your wife or in its people is a or a plus and take half

Vikash Ranjan · Answer

let here. Come on, B be fixed Point and first quadrant on DDE. It&#x27;s off. D isn&#x27;t some off distances from Did you know to Joe DiMaggio? You&#x27;re gonna be on DDE. It&#x27;s minus. So first of all, we&#x27;re soul, but a of the question So But the stairs If there is lesson Joe there in distance from D comma Jiro two other three by its can be they&#x27;re used by increasing day. If it is here that a then distance from did God Zito to Father three. Why was he can be They&#x27;re used by the big day. No minimum. It&#x27;s off, Dean. A curse for in this on Lee. So someday is equal to D plus twice off Demonize a horse Scott Glass Be a squad. So this is is Dennis and Dee task that it isn&#x27;t? Dusty is a first delivered. The best T is equal to one class myself. Dumanis is a bomb d minus a whole squad Plus So for political points we will have to put s tastic. But zero, this is equal. Do one bless toe D minus a support will go off demons. A police car. Last so, after all, refined the equals a minus. B A route to D. This is our be on DDE. No, the question father is if Big is less than three, then indeed is written Geo on DDE. I mean us at the at. If this is your little tree, eh? Minimum of us at the great deal. Part B off the question here is this is the X and y axis. So this is the we&#x27;re off for. There&#x27;s one of you b equals 0.5 on DDE. This is the graph for B equals three and this is a good off for being request three. So from this, we can see that minimum our cars for the equals one minus 0.5 over. Photo treatises equal to Europe on seven Killer reminiscent from Graf Won And the minimum. Of course, at the request of you, this is confirmed from Graff through and see from. But after 11 No, they will find its off d. This is equally deep, less to ourselves. B is far less a minus. D Holy Squire for the zero a gross opponent be bust three There four years off it. This becomes your plus. Try LeVar man. Plus one. This is Eagle Do No. I don&#x27;t pretend this is equal to 6.3. Do for five. This is the answer to the given problem.

Ian Shi · Answer

all right, it&#x27;s going to be constructing United seven. That&#x27;s three quarters the length of a B to do this. Recommended buying this into two permanent court by sectors. So it&#x27;s over the compass. Maybe we&#x27;ll per particularly bisect. This were just bisected. Doesn&#x27;t happy Particular was gonna find the point that it goes through. It&#x27;s a little dryer. Curves like So get our bisecting point. That wasn&#x27;t really good drawn, very quality line. There we go. That&#x27;s bisected, and I will do it again. Same process. But for this point just did and be do, uh, glad down. And there&#x27;s our second bisected put. So without saying, is that this distance right over here three force, maybe.

Suman Saurav Thakur · Answer

and and it is given there to see is the point online segment A B. That is twice as far be from. Okay. So if this is a line segment A B and C. Is the point. Okay, C. Is the point? Um that is twice as far be as it is from. Mm. So if this is C. Okay, And if this is X, this becomes two x. And given that small is oh, a vector small, B. S O. B victor, and C. Is the victor? Okay. Uh huh. Okay. If this is oh so a vector is a then we will be victorious. B. Okay. And we&#x27;ll see victories. See selected, going to show that sequel to two or three A plus one by three. B. Yeah. Okay, so if this is like this, so a B victor If A B. S. X. So this is one by three, this is okay. And see victor be returned evicted this, you see. So here, if we write it as a C. Like this, so in the strangle we can write a vector equal to sorry, evicted place X. Vector or vector will be equal to see victor. Similarly here we can right see vector plus expect so I do expect equal to ah be evicted. If we could just eliminate the value of facts, we have to multiply this with two. So this becomes to a vector Plus two Expected equal to to see Victor and see victor place to expect. It will be equal to be evicted. Now subtract minus minus minus this will get cancelled out. So we should right here as to a vector minus c vector equal to to see vector minus b victor. So if we add selected to both, the sides will be getting three C vector equal to be vector plus. Do it victor. So let&#x27;s divide both sides by three will be getting see victory equal to two by three A vector place one x 3 B victor. This is being written here as a sequel to to buy three a Plus one x 3 B. Since Fruit. Thank you.

Charles Carter · Answer

Okay, So, uh, information gave us is that a is less than B and they&#x27;re both positive. Yes, it for a since the both positive doesn&#x27;t really matter about the less than thing we&#x27;re in Quadra blind, since both values are positive, Uh, for be that we have a negative a and positive beam that&#x27;s gonna be in quadrant Teoh. Okay. See, B minus a. And since B is greater than a, that will be positive. So we have a positive first value in a negative second value, which will put us in quadrant for and the A minus bi will be negative. And then negative B minus A will just be more negative. So these are both negatives. That&#x27;s gonna be in quadrant three. Thank you very much.

Amrita Bhasin · Answer

The problem is let A and B B positive numbers find it the laugh of the sharpest the line segment that is cut off as a first order in&#x27;t and the passage through the point of a B First we write down the equation of this line. This is ax over you Us. Why over me? It&#x27;s equal to what? So we have this part. It&#x27;s the culture. You This part is to win like how to be the laugh off this segment half now square is what you use Quiet. Plus we square things. This line has been through the point being We have a over you ask the overbearing Is that what you want? From here? We have a B over three is equal to one month they have over you Is that this is you once, eh? Over you. So we have me t going to be you over you months, eh? Then how square it is about you, you squire Plus you, squire, you square you once a square So we want to find the minimal value off our So rest a nightstick derogative i&#x27;ll square This is culture to you plus b square hams. You you, you money, eh, Squire? One of the few squared hams. You must a house too. Over. You might stay to the part ofthe war. We can cancel out. You might stay here. Three. That&#x27;s this is culture to you. Pam&#x27;s one minus a squire over you minus a the power of three. If we liked our pro outscore promise you got zero. We have. You must be killed if they called you a square. So have you. It&#x27;s the culture. Hey, you want third house speech? You&#x27;re to third, eh? And his council? You coach you a To the power of my on speed The power of third us, huh? To the papa too. If you is greatest on this number, we have now square prom is which is on zero. And if you is the last time this number we have out front I&#x27;ll score from his less than zero. So when you is equal to this number, ask where has a minimal value? How many moments, Squire? This is a culture just plugging his number two disfunction over half this&#x27;s equal chul a pop up to third. Asked he to the path ofthe to third score. We&#x27;re half AL minimum. It&#x27;s cultural A to the power off to third as being to the power of two there.

Amrita Bhasin · Answer

were given to positive numbers A and B, and we have to find the length of the shortest line segment that is cut off by the first quadrant and passes through the point a. B Just get one. Well, first of all, equation of a line that passes through the point A B with a slope of em is why minus B equals M times X minus a. Now it&#x27;s intuitively mm. Has to be less than zero. Otherwise, if we had a positive slope, there would not be a line segments cut off by the first quadrant. Now set X equal to zero. We can find the coordinates of the Y intercept. So we have a Y intercept with coordinates zero mhm negative AM plus B and an X intercept. You said Y equals zero, and we have that X is equal to negative B over M plus A. This is negative. Be over em, plus a zero. Yes. Therefore, the distance from the Y intercept to the X intercept shall call D as a function of the slope M. This is the square root of the difference of the X components, which is a minus B over M squared plus B minus AM squared. Now, of course, we know that she our function d is minimized at the same time when it&#x27;s square is minimized, I&#x27;ll call this function F. So F is a function of em is a minus B over em squared plus B minus A m squared. Now, to find the minimum of yes will differentiate and find where this is equal to zero. So f prime of em is two times a minus B over em times derivative of the inside. This is positive. Be over m squared plus two times B minus. I am in Pampers No times the derivative of the inside, which is negative A A. Now, if you simplify, we get to over M cubed times a B M minus B squared lawyers, plus a squared them to the fourth minus a b m cubed. We set this equal to zero. We can then factor by grouping so we get to over m cubed times and then we factor b out of the 1st and 2nd terms. We have B times a M minus B and we factor and a M cubed out of the second and 33rd and fourth terms. We get a minus. Yes, B sorry. A M minus b again, Just it is equals zero. I see it&#x27;s going to be factored as to over m cubed times B plus A M cubed times a M minus B equals zero. And with this factory ization, well, as you&#x27;ve already pointed out, Okay, One possibility is that a M minus B equals zero. This would tell us that M equals B over a, which is positive. So this is not the answer. As we already pointed out, the slope needs to be negative. Therefore, the only solution is the other factor. B plus a m cubed equals zero. And so M is equal to so mhm negative cube root of B over a. Yeah. Now, in fact, a few think about the way we factored f prime of em. It follows that f prime of them. Mhm is because I think less than zero. If em is less than negative. Cuba to be over a and f prime of them is greater than zero. If m is greater than negative Cuba, it&#x27;d be over a. Therefore it follows that f has a minimum at M equals negative cube root of B over a. Yeah, I will plug this back into our distance function, so we have f of negative cube root of B over A. This is equal to once you finally simplify it down. There&#x27;s a lot of steps here. I&#x27;m skipping. Mhm, a squared plus three a to the four thirds times B to the two thirds plus three. Eight the two thirds times B to the four thirds All right plus B squared. That&#x27;s so I&#x27;m skipping some steps here, my peeps. But then the length is the square root of this. Now to simplify this, that we can actually factor this as yeah a to the two thirds plus B to the two thirds cubed, just huge me boy and therefore the distance a d of negative cube root of be over a is the square root of this, which is a to the two thirds plus B to the two thirds to the three halves. Power that. And so this is our shortest length

Problem

At which points on the curve $ y = 1 + 40x^3 - 3x…

Problem 55 Hard Difficulty

Let $ a $ and $ b $ be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $ (a, b) $.

Answer

shortest length is $\sqrt{a^{2}+3 a^{4 / 3} b^{2 / 3}+3 a^{2 / 3} b^{4 / 3}+b^{2}}=\left(a^{2 / 3}+b^{2 / 3}\right)^{3 / 2}$

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shortest length is $\sqrt{a^{2}+3 a^{4 / 3} b^{2 / 3}+3 a^{2 / 3} b^{4 / 3}+b^{2}}=\left(a^{2 / 3}+b^{2 / 3}\right)^{3 / 2}$

Calculus

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James StewartCalculus

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Calculus