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Asoliad Iies barteen planas perpendcular {@ Ihe X-axis a1 * = and > = 4 The cross-secbons perpendkcular lo the K-aris betheen these plznes ar0 squares whosa bese...

Question

Asoliad Iies barteen planas perpendcular {@ Ihe X-axis a1 * = and > = 4 The cross-secbons perpendkcular lo the K-aris betheen these plznes ar0 squares whosa beses run bom Iha semicicle y = 416-* Ilw semicicle y = /16 Find tte rolume o/ Iha goldtolumx of ule sold @5 cublc Unrs (Stmetly Your ans+9t )

Asoliad Iies barteen planas perpendcular {@ Ihe X-axis a1 * = and > = 4 The cross-secbons perpendkcular lo the K-aris betheen these plznes ar0 squares whosa beses run bom Iha semicicle y = 416-* Ilw semicicle y = /16 Find tte rolume o/ Iha gold tolumx of ule sold @5 cublc Unrs (Stmetly Your ans+9t )



Answers

graph each semiellipse. $$y=-\sqrt{16-4 x^{2}}$$

For this problem, we have been given the equation of an Ellipse X squared over 16 plus y squared over four equals one. And our goal is to sketch this ellipse. Now, in order to do that, let's take a step back and review the standard form for an ellipse so that we understand what all the numbers are in. Our equation on the lips is set up as two fractions. I'm going to write the numerator for the moment. I'm gonna come back to the denominators in a minute. My numerator r x minus h squared Plus why minus k squared. And when we add these fractions, they're going to equal one. So those numerator should look very familiar. It looks very similar to the standard form of a circle. And just like a circle, the center of my lips is going to be the point. H k, Let's take a look at our equation. In this case, H and K are both zero. I'm not subtracting anything from X and y so the center of my lips will be the origin. The 0.0 Now let's go back and look at those denominators. The biggest denominator we did note with a squared. If that biggest number is under my ex term, that means excess my major axis and I have a horizontal ellipse. If I switch these denominators and I put the bigger number under the Y term, that means why is my major axis and I have Ah, horror or I start I have a vertical ellipse. So where the bigger denominator is tells me the orientation of my lips. Well, if I look at what we've been given, the biggest denominator is under the X term. So my major axis is along the X axis and a square to 16, which means a is four. So I'm going to start at the center. On my graph, I'm gonna go along the major axis four units in either direction those air, the vergis ease of my lips. Now how wide or how skinny is my lips will To find that out, we need be, we go to the other denominator, which tells me that B is too. So I'm going to go up two units and down two units on my minor axis and then connect my dots that will give me the sketch of the Ellipse equation that we were given

When we graph the semi ellipse which is already solved for why we see, we have X intercepts at negative one and one and our Y intercept is at zero comma. Negative too.

For this problem, we have been given the equation of an Ellipse X squared over four plus y squared over 16 equals one and we want to graph this ellipse. Now, to do that, let's take a step back and look at this standard form for an ellipse. So we understand what all these numbers mean Now we're gonna have to fractions. I'm going to start with the numerator is first all right, the denominators in in a moment. My enumerators are X minus h squared. Plus why minus k squared and this adding them together is going to give me one. Now those numerator should look very familiar. Those are very similar to the standard form of a circle. And just like in a circle, the center of my lips is going to be the point HK. So if I look at my given equation here for look at those numerator both h and K R zero, I'm not subtracting anything from X and y so the center of my lips is going to be at the origin. Okay, now let's look at her denominators. We use a to show the biggest number in the denominator. If the biggest denominator is under X. That means the X is my major axis and I'm going toe have ah, horizontal ellipse. I'm gonna race these denominators. What if they're switched? Let's say my biggest term is under the why term or my biggest denominators under my Y term? That means that why is now my major axis and I'm going to have a vertical ellipse. So those denominators tell me the orientation of my lips. Now let's take a look at the Ellipse we were given. The biggest denominator is under the why term? So my major axis is the Y axis and A is the square root of 16 or four. So I'm going to start at that center 0.0 going to go along my major y axis, four units in either direction. Thes are the vergis Ease off my ellipse. Those of the outermost points on the major axis. Now how fat Or how skinny is this lips to find that out? We look at the minor axis under the smaller of the denominators That tells me that B is too so on my lips. I'm going to go out to units on the minor axis from the center going either direction. And now I'm going to connect thes four points into a sketch of an Ellipse. So that is the sketch of the Ellipse for are given equation.

We are going to do prolong number 19 in discussion. We need to just sketch the graph of The given function that is y equals two Under Road of X. Men is four. So the matter which we will use is grab utility method in this matter. We just put random values of facts and we find the value of Y. And we applaud that. And the plane to get the graph. Now we are going to keep X as a zero. Then we will have Y. S minus four. Now when we are going to keep X as let's say two then how this will be let's keep it a number which will give us a perfect number as Y. And then that will be easy for us to so so if you keep access for Then this will be to manage four. So that has managed to Now if you are going to keep y. s. four. Okay, uh Y. S. Four then this will be a Y. S. Foreign. Later skip at Y zero. We can keep any value that we want. Okay, so this is we are keeping white as Jiro then at the value of access coming out to be For school. That is four square. That is 16. Okay, so we're going to just look at this point on the plane. So this is why you access this is a uh minus Y axis xxx supposed to access management success. He says, let's say 11 is four. So this is eight, This is 12. This is 16. Same follows for negative side. Manus for Manis said managed to man that's 16. No sam phylloxera access. See I'm full of so negative X access. And where does it start making the point? First point is X zero, Y is minus four. It's access, you know? And why is -4. So this is the point. No, when access for why is managed to. So this is another point now in excess ah 16. Why is you know? So this is the another point. So we got the graph, that graph will be going something like this. Okay To do touching at 16. So let me just make it clearly, this is how we drop. It looks like we can take any scale that we want. I have taken here when you know test phones, so you can take one unit as one. Also, if you have ah very spacey graph is basically an Okay, so this is what we got now writing acts interceptor. So actually intercept 2016. Okay, as the graph got carsick success at 16 here And why exist? Why interceptors -4? So this is why intercept This is -4. So this is the answer in this case. That's all. Thank you.


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