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[Maximum mark: 15]Consider a triangle OAB such that 0 has coordinates (0 0 ,0),A has coordinates (0 1,2) and B has coordinates (2b , 0 b _ 1) where b < 0 _(a) Fi...

Question

[Maximum mark: 15]Consider a triangle OAB such that 0 has coordinates (0 0 ,0),A has coordinates (0 1,2) and B has coordinates (2b , 0 b _ 1) where b < 0 _(a) Find, in terms of b, a Cartesian equation of the plane II containing this triangle.Let M be the midpoint of the line segment [OB] -(b) Find, in terms of b , the equation of the line L which passes through M and is perpendicular to the plane II.(c) Show that L does not intersect the Y-axis for any negative value of b _

[Maximum mark: 15] Consider a triangle OAB such that 0 has coordinates (0 0 ,0),A has coordinates (0 1,2) and B has coordinates (2b , 0 b _ 1) where b < 0 _ (a) Find, in terms of b, a Cartesian equation of the plane II containing this triangle. Let M be the midpoint of the line segment [OB] - (b) Find, in terms of b , the equation of the line L which passes through M and is perpendicular to the plane II. (c) Show that L does not intersect the Y-axis for any negative value of b _



Answers

$y \leq-15 x+3000$
$y \leq 5 x$
In the $x y$ -plane, if a point with cordinates $(a, b)$ lies
in the solution set of the system of inequalities above,
what is the maximum possible value of $b$ ?

So for Question 42 we wanna has two parts. Some party. We want to find the Max Valley of W on the line of intersection of these two planes. So those three planes are gonna be my constraint sets. What's constrained this function? What I'm trying to maximize. And so they're gonna be my g one of g two's gonna have G one equal X plus y. Plus is the minus 40 zero energy to is gonna be X plus y minus. Izzy's equals zero. And so what I'm going for it is I am will be using equation to from the book. And that tells me that the greeting of s is equal to land the time so grating everyone plus, um, you times ingredient of G two. So I'm looking for the X y and Z values that satisfy this equation. So that means my first up is I do need to find ingredients of these three equations. So for the grating f taking the partial derivative with respect to X, I get y Z. I was respected. Why? I get ecstasy and with respect, Izzie, I get X y taking the radiant of G one. I see that all of the partial derivatives or just unequal mind something have I plus A J Plus K and then for the gravy Aunt Aggie to I'll have again I plus J in this time of minus cave. So I'm went grating about equal to land a Time Street one in mu time. She chews on Go ahead and multiply out my lambda and in my Mu. So this question really just means that my coefficients of eyes that would have y z equal to land a plus mu my coefficients of Jae's exes. He's been equal to land a plus mu, and for my case, ex wives been equal to Lambda minus from you. So again, Ryan, that out, I have y Z sequel to Lambda Plus Mu X Z is equal to Lambda plus mu, and then X Y is equal to lambda minus you. So right off I noticed that Y, z and Exeter both equal to the same thing. So that means that Y Z is equal to X Z. And I see that this has two different solutions. Either Z's been equal zero or why is evil ex so the only see that are going to satisfy that. So that means I'm gonna have two different cases here. So starting with case one when have Z is equal to zero? So was the musical zero than looking back at my G functions. That means the experts wise Eagle 40 and experts why this equals zero. However, this cannot happen. There's no way the experts why he can both equal zero and 40 Asian summer move on the case to you were X is equal to why so going back to my G equations That means I want to start with The two accesses are both equal the same thing. So too explicit, easy below 40 and then two x minus. He's equal to zero. And so when you saw this, you should get that X is then equal to 10. That means that Y is equal to thanks. They're both thinking the same thing and then plugging in 10 for X, you should get that Z is equal to 20. So, Mr Max value is gonna happen at these values for X, y and Z So plugging those in I get that w of 10 10 and 20 see also the product of those which is equal a few 1000. So next in part B, it tells us to give a geometric argument to support that we found in Max and not a men value of W. So you have a geometric argument and we'll look at the normal vector between these two. So it's found was determinant. Hi. Okay, I say the coefficients from the tops. That's 11111 negative one. And this comes out to negative Thio I plus to J. And so I know this is parallel to the line of intersection and so therefore, our line is X is equal to negative to t plus the value found for ex earlier was just 10. Why is equal to two tea plus the Wye Valley found pretend and then Z is equal to There's no Katrine here sit zero t or zero. And then we found before that it was 20. So that means the w, which is equal to X y Z, is not gonna be equal to negative to t plus 10 times duty plus 10 times 20 which when you multiply that out, you should get negative. 40 square plus 100 comes funny and I see this toe has a maximum tease equal zero because that would give us some 100 times 20. And so therefore, it is what we have found. A maximum and not a minimum.

Um So we're given a plain X plus three wide plus the process on a rectangle are with Venice's wow 00 A couple zero So it could be and a comma be We know that a copy lies on intersection Yes, Off express three wide We'll see equal six and x y plane. What this means is that yeah, this means that I'm a plus three b most equal six All right, I'm weed. We're interested, right? And the thought of a baby for which for you the following the volume Also the solid between expressed three y z equals six and are its maximum not to start we first. Well, we compute computing does vital on z d a. All right now we know that to see, um is even equal. Six months X medals three wide just by right model over here. And I we wanted our region are equal or ex common. Why? For which there is unequaled X less than equal a equals. Why little equally be eso? This region here seems having integral the y from zero to be integral over x from zero to a Uh huh six minutes x minutes three wide so dx dy white on this equals integral from zero to p off six Xmas half X squared that is three x y from being a 50 y, which equals integral from zero to be off. Six. A man is half a squared man is t a Y de y. Then there's equals. Um six. A. Y manus half a squid y. Then there's three to a Y squared from zero B, which gives us six. A B man is half a squid, the minus three of the 28 Disquiet. Now, to make it easier to simplify, we use the fact that a plus three B or six, which basically implies a most equal six that is three d. So put that into expression. Here we have that just part with equal six into six minutes. Three B mammals would have into six minutes. Three B squared times. Be ministry developer to six Ministry. Be this quit and there's the equals. Six. Oh, just be here. There's an equal 16 to 6 b ministry B squared that is one half into 36 minus 36 B plus 90 squared top speed minus three halves into six. Disquiet managed three b. Cuba Much Onda simplifies Thio, um 36 B minus 18 is quiet minus 36 over to last 36 b over to Manus Man of the two Does the baby here a piece called here and it's not double team. Be cute. Minus 3 to 6 is quiet. Class three of the two three be cute, Which gives us no citizens. Be man is 18 b squared minus 18. The last 18 discord the minus nine house Be cute. Minus name is cleared. Plus, I'm over to the hint pretty equals hated be eight and G thinks 18 and be this to cancel this to cancer And then we have a negative night. He's quiet. Oh, So what this means is that, um yeah, double integral over our off the d A. Equals eight and B minus nine p squared. Let's just call it some VP. So we have written the double integral or double integral of the D A in terms off the But remember, we want, um, you want a baby for which double integral are of Z D. A. Is Max. Okay? We're gonna apply some principles from calculus from different calculus um um, Speed, we find you find the primal be equals zero solve for being so the prime of B equals of 18 minutes 18 p equals zero. We didn't get off 18 equal. 18 be impacted people. All right. Also, we do the conservative test. The program off B equals negative 18 which is less than zero. So we know that the equal one maximizes 50. Um, using the fact using the relation that six equals a plus Street B, we have a six with equal a plus three into one, which equals a plus three that present a equals six ministry, which equals three. So what that means is that the 0.3 come award maximize Integral are the d A. So the answer is long. Three. Come over. There you go.

Were given a teacher here, um, with together in volume and were asked to find a minimum volume among all the planes passing through the point 111 This tetrahedron is formed from the plane with equation X over a plus. Whatever B plus C ris equals one and the positive coordinate planes and it has a volume. V equals 1/6 a times B times C In this tetrahedron is pictured in figure 20 following exercise. Are you five of this section? Now we know that claim was passed through the 0.111 So the point must satisfy the equation of the plane. Must have that one over a plus one Overbey plus whenever C equals one. So we want to minimize the function. The of ABC equals 1/6 a b c. Subject to the constraint G of ABC equals one over a plus one. Overbey plus one oversee minus one equals zero. When we have that as we're needed first Oct int A, B and C must all be greater than zero. Do this. Let's write out the garage equations. We have ingredients of V is the victor 16 BC 16 a C and 16 AP Grady int of G is the vector negative one over a squared negative one over B squared negative one over C squared and the LaGrange condition ingredient of V is equal to Lambda Times. Ingredient of G gives us thes LaGrange equations 1/6 B C is equal to negative one over a squared 1/6 a. C Sorry times lambda. 1/6 of a C is equal to negative one over B squared lambda. 1/6 of a B is equal to negative one of her C squared times lambda were asked to solve for Lambda in terms of aid me and see Now our equations imply that Lambda is equal to well, we have negative a square to B C over six, but we have that Lambda is also equal to negative. A C B squared over six in the Lambda is equal to negative. A sorry A B C. Squared over six. So now want to solve for A B and C using our constrained equation. So, quitting these expressions for Lambda we get that BC a squared is equal to a see B squared, which implies that a Times B times C times a minus bi equals zero. Likewise, we have dead a times B times C squared is also equal to a C B squared and so we have that a times B times C times C minus B is equal to zero. Now A, B and C are positive numbers. Therefore, that is your product property. It follows that a must be equal to be and that he must be equal to see. So all three of a B and C are the same. I was substitute into our constraint. So we have one over B plus one Overbey plus one over B equals one. Or in other words, three Overbey equals one, which means that B is equal to three, therefore follows that A and C are also three since they're both equal to be. And so we obtain the critical 0.3 three three. Now we know that if V has a minimum value subject constraint and it occurs at this point, so the plane that minimizes be would be X over three plus why over three plus z over three equals one four x plus y plus Z equals three now and we still have to show that the has, um um value subject constraint. However, a constraint which is a plane is not bounded. So we need to justify the existence of a minimum value of our volume under this constraint. Well, again, we have that a b and C or non negative. And we also have that some of their super closes one from this. It follows that none of them contend to zero. Otherwise, you have infinity in the left side and one on the right side. And in fact we could be even more specific. We have that A, B and C must all be greater than or equal to one. Nothing could be less than one. Otherwise we'd have a term greater than one on the left side and one on the right side. And so it follows that the volume the must approach infinity has a approaches infinity or be approaches infinity or C approaches infinity. This means that you can find a cube with the point one third one third, one third such that on the part of the constraint outside of our cube, our volume the will be greater than our volume at this 0.1 3rd one third one third, which is equal to 1/6 times one third cube, which is 162nd. Now, on the part of the constraint inside the Cube, we know that our volume V has a minimum value. We'll call em because this part, which is inside the Cube, is closed and bounded in our three. Clearly, it follows that M is the minimum of the on the whole constraint putting these two statements together. Therefore we have the minimum volume is 100 one over 162.

Okay, so we're given the maximum on the line of intersection W equals X Y Z. We have our plane X plus y plus Z minus 40 is equal to zero, and the Sphere X plus y minus Z is equal to zero. So if we let G one of X y Z is equal to X plus y plus Z minus 40 equals zero and G two of X y Z equals X plus y minus Z equals zero. Then if we take the Grady int of G ones that hit the partial derivative here with respect to X off X plus y plus Z minus 40 times I plus the partial respect the why of the same thing Times J plus the partial with respect to Z off that same thing, Times K. Okay, um well, that's equal to I plus J plus K. So Therefore, the Grady int of G one is I plus J plus K and the Grady int of G to like what we take. The partial now is back. The X off each component and see the ingredient off G two is equal to I plus J minus k. Okay, so now it is given that w of x y Z is equal to X y z So therefore, the Grady int of w is equal to the partial suspect The x of X y z times I plus the partial perspective Why of X y z times j plus the partial with respect to K of X y z times K which is equal to y Z I plus ex zj plus x y que Okay, um, so the Grady int of W is equal to Lambda Times, ingredient of G one plus mu times ingredient of G two. Okay, this is gonna apply. Imply that, um Why is equal to X or Z is equal to zero. So we then have to cases toe Look at, um, case one if Z is equal to zero, but then we have g one is equal to or x plus y plus zero minus 40. That's just equals X plus y, which is equal to 40. And, um, G two. We get X plus y is equal to zero. So if we call this equation one in this equation to while solving one and two, we end up with no solution. Okay, The second case would be if x equal to y. So if X is equal, why, then we have, um G one is equal to X plus y plus Z minus 40 equals zero. So we get X equal wise, we get two X plus Z minus 40 equals zero. Therefore, two x plus zzz could've 40 and g to Well, we end up with two X. Um minus Z is equal to zero. So we solve these two equations here and get that Z is equal to 20. Okay, um, so we end up so we get that X equal to 10 on why is he go to 10? So therefore we get that w is equal to 10 times 10 times 20 which is equal to 2000. Okay, So now, looking at part beat well, we're given that w is equal to X y Z and G one is equal to X plus y plus Z minus 40 equals zero, and G two is equal to X plus y minus Z is equal to zero. So from G one and G two, we get that end is equal to the determinant here of this matrix which, um, is equal to negative two I plus two j, which is parallel to the line of intersection. So the line is access equal to negative to t plus 10. Why is equal to two t plus 10 and Z is equal to 20. So we get that W um which is equal to X y z, which is equal to negative. 40 squared plus 20 t minus 20 T plus 100 all times 20. So with the W is equal to negative 80 t squared plus 2000. And this has a maximum when t is equal to zero, right? Why? Well, this because I mean T square is always gonna be positive, multiplying by a negative coefficient out in front. This is always gonna be negative. So does smallest. I mean, when we're adding 2000 to this, so the maximum value would have to be when t is equal to zero. Right. Um so therefore, we get that access equal to 10. Why is equal to 10 and that Z is equal to 20 which is what we found in part A. So s Oh, yeah, that shows it. And we're done. Take care


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