We are given the temperature in three space as a function T of X, Y. And Z. And were given different values of temperature T. And were asked to draw Aisa therms corresponding to these temperatures. So the temperature injury spaces T of X, Y Z equals X squared over four Plus y squared over nine plus Z squared Okay. Okay. Mhm. Yeah. Now to draw these ISA Therms will simply plug in different values of the temperature and then grab the corresponding equation in three space. So just for example, if T equals zero, Well, this gives us zero equals x squared over four plus Y squared over nine plus Z squared. We see that in fact all of these ice with terms they're going to be ellipses Or now ellipses but ellipse sides since there in three space except for T equals zero. In this case we can solve for Z. And we get that Z is equal to plus or minus the square root of negative x squared over four plus y squared overnight. Thank you this company. And this is the equation of a cone. Mhm. Yeah. So we have our three axes X, Y&Z. Yeah. Mhm. Mhm. Now, as a matter of fact, this is a very special kind of cone because we see there's only really one solution to this equation which is the point X, Y and Z are all zero. And so this is the cone which is really just the origin. And therefore our graph is a single point shall draw in red the point of the origin. So there's only 1.3 space which has a temperature of zero. That's what the ice of Durham tells us. Now when T equals one, I'll draw the resulting graph in green. Well then the equation becomes one equals x squared over four Plus why squared over nine plus z squared. And they see that this is the equation of an ellipse oid centered at the origin. So we draw X, Y and z axes. Yeah. Now along the X axis we have a radius of Square to four or 2. Yeah. And so we have points here and here along the y axis we have The radius of three. So we have points, we have points here and here as well. And then finally along the Z axis we simply have a radius of one. So we have points at 001 and 00 -1 connecting these points. We obtained the ellipsis. Mhm. Mhm. Yeah. And so the ellipse oid looks a bit like this. It's kind of a rough drawing but this makes a little bit clearer and you sort of can see it now. Yeah. Okay. Uh huh. And finally we'll draw the ice of term corresponding to T equals four. And I'll draw this in blue. So this gives us the equation four equals x squared over four plus Y squared over nine plus Z squared Dividing both sides by four. We get the standard form, one equals x squared over 16 Plus y squared over 36 plus Z squared over four. And in this form it's easy to see it. This is lips oid centered at the origin. Mhm. Now, from our equation we see that we have a radius of four along the X axis. So we have points at 400 and negative 400 is located about here and here and then along the Y axis. Sorry this isn't blue, not green. We have a radius belt, six along the y axis. We have points at 060 and zero negative 60 Mhm. So these are located about here and here. Finally, we have a radius to along the Z axis. So we have points at 002 and 00 negative two. But you're located about here and here, connecting these six points we can obtain the graft part looks oid okay, yeah. And the graph of the ice a therm looks something like this. This is an approximation of the flip side and the size of term represents all the points in three space where the temperature has The measure of four.