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Consider set P described by linear inequalities constraints, that is P = {" € Rn ai < bi,iPage 2 of 3m}_ ball with center y and radius is defined as ...

Question

Consider set P described by linear inequalities constraints, that is P = {" € Rn ai < bi,iPage 2 of 3m}_ ball with center y and radius is defined as the set of all points within (Euclidean) distance from y. We are interested in finding ball with largest possible radius, which is entirely contained within the set P_ Provide a linear programming formulation of the problem_

Consider set P described by linear inequalities constraints, that is P = {" € Rn ai < bi,i Page 2 of 3 m}_ ball with center y and radius is defined as the set of all points within (Euclidean) distance from y. We are interested in finding ball with largest possible radius, which is entirely contained within the set P_ Provide a linear programming formulation of the problem_



Answers

(a) Find inequalities that describe a hollow ball with diameter
30 $\mathrm{cm}$ and thickness 0.5 $\mathrm{cm} .$ Explain how you have posi-
tioned the coordinate system that you have chosen.
(b) Suppose the ball is cut in half. Write inequalities that
describe one of the halves.

The problem is finding inequalities that describe a hard war by Stanley to thirty centimeters and thickness there. Point of far off on the meters. Explain how you have position or coordinate system that you have chosen part A if representation. The center of the boy ridden off X Y Z coordinate system. Sorry, The ball with Star May to thirty seven meters can be described as X Square plus y square. I asked a squire, which is the last time fifteen square to get the sickness there upon five twenty meters. We need to take house. Sorry, war with time later. Thirty minus zero point five house, too, which is equal to twenty nine centimeters, which can be described as X Square plus y square in your corner, which is the last time. Fourteen point five square. So a hollow ball on be described as fourteen point five square. It's the last time X corner y score here. C Square, which is last sound of teen score Hot B. Suppose the board is cut in half right in you qualities that describe one of the huffs. So you think katydids war through the X Y plain, then the upper half and be described pies. C is grits and zero on DH X squared plus y squared plus Z squared. It's between fourteen point five square and on fourteen. Score. Fifteen square it's the upper off.

Okay, so we have a couple of inequalities. So we have, Um the 1st 1 is a hollow ball with, um So I write down how the ball hollow ball with diameter of 30 centimeters and thickness of 0.5 centimeters. So, um, right, this, um, in the Tootie's place first. So Well, I'll write a circle. Um, and then we converted to spherical. So we're in a circle. We know that we want, um, the diameter to be 30 swift 30 across. And it's gonna be a thickness of 0.5. So we want thickness to just be this outer part. So I have this outer part here, and we know that that's 0.5. So perverting this to a spirit kal important. Since it's the hollow ball, it's gonna spend the whole, um ah, data and five space so we can write it. But we don't have to. Um, but we can just right. The following is just described by since we know our diameter must be 30. We know our Max radius or Max row just 15 and then our thickness in out of this case corresponds to a You don't have to actually cut that in half because it's all around. So then this is just still 0.5 centimeters. So we can actually describe this entire ball using the inequality. Ro is in between 15 and 14.5 centimeters. So then what we like to do now is, um, right inequalities that described half of this. So we're going to say we cut it half way vertically. So we cut this vertically. Um, we end up with just this half. We'll go ahead and say, This is, um positive X and y. So we have Let's say this is positive X. So, um, we can go ahead and use our cornet space who describe a this hollow ball. We'll say it's right better. Who will say it's It's in this this plane So positive x in the positive X direction. So, um, by since fires angle with Z axis, it's still encompasses the entire identify, so we don't actually have to write that. And then we know it's different in x and y because data is different. So data ends up spanning negative Pirata, pirate or two. So this isn't negative to this estate. I see quite a negative. Prior to this line and right here this line is Paradies. You could deposit pirate too, so we can go ahead and say for our our half of the ball will say that this is described by these two inequalities the same as before With our radius that describes our actual sphere and the cut it in half. We'll show that how data must be between ah, pyre or two and native prior to And pirates too. And in for both cases. If you wanted to write that, you could, um if I just spanning from zero to pi And in the other case, um, data goes from zero to part 20 to 2 pi. So, um, to be as specific as possible, all you have to do is write these two qualities.

In this problem. We plotted this craft using the this constraint and the graph has given you in the question. Okay the point a 00. This 0.40 is the X intercept of the equation. To and at this point C. 34 either intersection point of these two equations and is fine by solving these two ago and simultaneously at this point D. Is the y intercept of this equation. one. Now we have the 446 of the physical region that we have to find where the the objective function is maximized and many miles. So five women plot point a. Values of point and objective function four point A. We have the equals two who into X. X. Value is zero hair plus five by and by is also zero. So we have the equal to zero. We will get clear four point B. And the values of X. And Y. Uh Z four and zero respectively. Supporting the values we will get he goes to add before that. Ad knuckle point C. He has X equals good three. And why calls through. Uh huh Forming it we will get the vehicles to 26. Finally the last one is B point their exes zero and they have wife occupied which when he was he was too 25. How you can see That we have zero value Here we have zero and Here is the maximum where you're 26. So that objective function is maximized at point C. 34 which gives us statistics and it is minimized at point a 00 which give us equal to zero. Thank you.

Okay in this problem we have to find a maximum and the minimum point. Further given objective from Ngos you go before expressed to you by according to the given constraints in the question. We have drawn the graph and we got the three into second points A B and C. So we will put all the values of a point A point B and point C. Separately in the objective function. First of all putting the value of four point A. In the objective Z. We will get Z. It was too four times zero. That's last three into zero because those are values of X and Y at point A R zero. And we will get the objective function er equal to zero. No going to the point B. The floor of the middle. Mhm We will disappoint me is five for the value of x is five and appoint me. And when you y zero point we will write it at four and to the five plus three and 20. And we will get G E close to According to five which is thank you. Last point it won't she? Similarly we we have to point C is yeah zero is the value of exit Quincy and find the venue of. Why our point C. And we will get reliable to 15. We can see that at point B. We have the maximum value 20 and our point we have the minimum value zero. So B is the maximum point B. 50 is the maximum point and a 00 is the minimum point. Thank you


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