Need Help?RdhKathkTkteaer0 05 points Previous Answers SCalc7 14 066Describe the level surfaces of the function. f(x, Y, 2) = x2 + 4y2 + 322The level surfaces are ...

Describe geometrically the level surfaces for the functions defined. $$f(x, y, z)=x^{2}+y^{2}+z^{2} ; k>0$$

For this problem, we are asked to describe geometrically the level surfaces of the function F of X, Y Z equals 100 X squared plus 16 Y squared plus 25 Z squared. So this uh this form should look rather familiar. Who I should add. We are also told that K is strictly greater than zero. So the form of this should look rather familiar. I'll write it out in terms of K, which means I could just erase F of X, Y. Z and replace it with a K. So this this should look familiar. If we divide both sides by K, we get one equals 100 over K X squared plus 16 over K Y squared Plus 25 over K. Z squared. So the reason that this should look familiar is that this is the this resembles the general form of a three dimensional generalization of the lips called an ellipse oid. So we would generally have X squared over a squared, Y squared over B squared, etcetera. So we would have that A is going to equal the square root of K over 100 B is going to equal the square root of K over 16 and C is going to equal the square root of K over 25 which we can reduce all of these down to root K over 10. Then route K over four And then root K over five. So that gives us the description that we have, essentially, we're going to have a lip sides with uh essentially the ratio of the axis is going to stay constant. If we imagine when K equals excuse me, when K equals one, then we'll have an ellipse oid that's going to be passing through one second here. So we have that We passed through the .1 over 10. Actually need to be careful. It's going to be for here, 1/10 and then 1/4 is a little bit further out and then 1/25. So very small. So we'll have and ellipse that's sort of very flattened. I can't really draw it very well in three D. But hopefully the idea is coming across. And then as we just k the the ellipse will be um scaled up larger and larger and larger, so the level surfaces will be these consecutively larger ellipses with the same overall proportions.

For this problem, we are asked to describe geometrically the level surfaces for the function F of x, Y Z equals 16 X squared plus 16 Y squared minus nine Z squared. So what we can do here is first of all, just replace F of X Y zed with a K. It's going to be our constant for a level surface. Then one way of approaching it uh sorry, If we divide both sides by K. Here and we'll end up with one equal 16 X squared over K plus 16 over K. Y squared minus nine over K. Z squared. Now, this equation that we have on the right hand side should look familiar because this is the pattern of a hyperbole Lloyd of one sheet. Specifically with a or uh it would be a square two equals K over 16. So A equals root K over four. The Equals root K over four and see equals root K over three. So what that tells us then is that we are going to have this hyperba Lloyd, She or sorry, hyperba Lloyd surface where the cross section and the ex wise r in the xy plane is going to be a perfect circle. And then as we go up in Z or up or down ins said, then we are going to have similarly varying value for the overall radius of this hyperba Lloyd. Uh Then what we're going to have as we are adjusting the value of K. You'll be getting hyperba lloyds of consecutively larger sides essentially enclosing each other. I know that's an awful sketch but I hope the idea is coming across here.

Right for this problem, we want to describe geometrically the level surfaces of the function F of X, Y Z equals E to the power of expert plus Y squared plus Z squared where K is greater than zero. So we can replace that F of X Y Z with a K. And now we have that X squared plus Y squared plus set square. That should look familiar. That looks like a sphere. So we want to get an equation for X squared plus Y squared plus Z squared. In terms of okay, we can do that by taking the natural law algorithm of both sides. So we'll end up getting that X squared plus Y squared plus Z squared. It's going to equal the natural algorithm of K. So what that's going to tell us is that we are essentially going to be dealing with um excuse me, brain part we are going to be dealing with for the level surfaces, spheres of increasing radius where the radius is going to be the square root of one of K. It is worth noting here that the there will be no meaningful level surfaces for K between zero and one, because expert plus Y squared plus X squared can never add up to something less than one. Or, excuse me, less than zero. And lawn of any number between zero and one will return something less than zero.

For this problem. We've been given a function in three variables X squared plus three y squared plus five z squared. And what we want to do for this problem is to find describe the level surfaces of this function. Now level surfaces of a function is we take this function and we said it equal to K some constant. So we want to know what values of X, Y and Z We can change them as much as we want. We want the whole functions value to remain constant, and hopefully we can change K two different constant values. And we can see what's happening to our function at different points. Um, it's a way of sometimes helping Thio visualize Ah three d shape a little bit more clearly. So let's see what we have here. Um, if I take this function that set equal to K, if I had a two dimensional function on, I'm just gonna take the x and y here. Let's say I had a two dimensional function equal decay where I had everything is positive and I have x squared and y squared. This is the equation of an ellipse. It's a little bit easier to see if we set it equal toe one like we would normally do for in the lips because that would give me X squared over K plus y squared three y squared over K. And if I want to put everything in the denominator like we often have feel y squared over B B squared, that would be my lips equation. Okay, so that's the two dimensional version. This three dimensional version is very similar again. Let me divide everything by K just you might be able to see this a little bit easier. X squared over K. That's why squared over K over three and I could do the same thing here. Z squared over K over five now. I wouldn't typically leave it with that fraction in the denominator, but it helps to see what's holding on with our lips. So this is an ellipse. Um, my biggest denominator is K K, which is under the X. So my major access for this ellipse oId this three dimensional lip shape is going to be along the X axis. And as I change k, this is going thio grow shrink, become different things. If I got if I put k equaling 20 then this kind of devolves into a single point right at the origin. As I get K bigger and bigger. My lips was going to expand, but these were going to be nested ellipse Lloyd's, um, as whenever I set them equal to K for these level surfaces.