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Parabotic reflector(a) The focal length of the (finite) paraboloid in the figure is the distance $p$ between its vertex and focus. Express $p$ in terms of $r$ and $...

Question

Parabotic reflector(a) The focal length of the (finite) paraboloid in the figure is the distance $p$ between its vertex and focus. Express $p$ in terms of $r$ and $h$(b) A reflector is to be constructed with a focal length of 10 feet and a depth of 5 feet. Find the radius of the reflector.(FIGURE CAN NOT COPY)

Parabotic reflector (a) The focal length of the (finite) paraboloid in the figure is the distance $p$ between its vertex and focus. Express $p$ in terms of $r$ and $h$ (b) A reflector is to be constructed with a focal length of 10 feet and a depth of 5 feet. Find the radius of the reflector. (FIGURE CAN NOT COPY)



Answers

A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is $ 10\;cm $.
(a) Find an equation of the parabola.
(b) Find the diameter of the opening $ | CD | $. $ 11\;cm $ from the vertex.

So we're told that we have a plane convex lens with the index of refraction in people the 1.55 and a focal length equal to 16.3 centimeters. It's positive, since it's convex and since it's a plain surface, it has an infinite radius of curvature. So let the plane surface be surfaced too. So are too. Then is going to be equal to infinity. We're gonna use Equation 23 Dutch 10 now to find the radius of curvature. Of the first lens are one so one over the focal length is equal to the index of refraction minus one, multiplied by one over R one plus one over R two. Well, we said that are too is equal to infinity. Break So one over infinity is essentially zero, so we can rewrite this as in minus one, multiplied by one over R. One where we want to solve for our one. Therefore, our one is equal to in minus one, multiplied by the Focal Inc So plugging the values in for the index of refraction and the focal length we find that this is equal to nine centimeters become box said it is the solution to our question

In problems. 62. We have this reflector dish, and the reflector portion of it is a cross section of a problem. Are given quite a bit of information here, and we need to figure out the distance wrong, essentially the vertex of the proble to the focus, which is that point F. And this isn't too difficult to do based off of some of the other problems that we've worked. So what weaken know is that this question mark this distance from the Vertex of the problems is gonna be Pete and we can get P. Um, if we set this up as a normal problem, I'm gonna draw a access here. That's gonna be why that's gonna be X. And then let's just go ahead and draw. But this problem would look like if we were just taking the cross section because we're concerned about this focus right here. And that doesn't change, no matter how much of the problem we're looking at. So once we have that, we know that. So the total length of this reflector dish is 20 feet, and then the vertex would be right in the middle. So that means that 10 feet over there's gonna be to the side of this reflector dish, and then we know that one foot up is the depth of the reflector dish. That's gonna be one foot. So that's our X and Y coordinate. So if it's 10 feet over one foot up, that's gonna be the X and Y value. So we can do is if we go to our equation for this problem. Now, I have it set up as the Y axis as the major access we're gonna end up having is we will have, um X squared is equal to four p. Y. And this is great. We have this pee right here, which is exactly what we need to solve in order to find the distance from the Vertex to focus. And we played in our X and Y values of X is gonna be 10. So it's gonna be 10 squared is going to be equal to four times p times. Or why value, which is just gonna be one. And I kind of wrote this kind of low. Let me move this up here, which is gonna end up giving us that 100 is gonna be equal to four p divide each side by four. We're gonna have P is equal to 100 and I decide by four p is not equal to 100. So I divided it. We're gonna actually end up having that P is equal to 25. So that is the grand revelation from this problem. Is that our distance from the Vertex? That focus is 25. The got a difficult portion of figuring out how to solve. This problem is just realizing that just because this is a parabolic cross section that doesn't change that the focus is gonna be the same if you're looking at the entire parabola or just that cross section so it can essentially ignore the cross section. And you just need that in order to find those X y values to plug in in order to sell it for this

So here we consider a parabolic reflector. Um, 400 z is equal two while x squared plus y squared. Okay, So for a parabolic reflector, the standard equation is well, instead, that equation is four p see is equal to x squared. Plus why squared? Okay, on the focal point is 00 p. So our focal point is zero zero. He and this would be our focal points points. Okay, so for us, the equation of the reflect Karabakh instructor is 400 z equals x plus y squared. So the variable Z has a linear value sold. The Z axis is the axis of symmetry. So the focal point lies on the Z axis. Now we compare the well provided equation here with the standard equation. Teoh get well, four, he is equal to well, 400. Okay, so therefore, P is equal to well, 400 before. Okay, Well, 44 4 equals well, 100. So therefore, P is 100 hens or therefore, the parabolic reflect. Their focal point is the point. So the focal point that we're looking for the focal 0.0 r um, local buoyant that we're looking for is, um is zero zero 100. All right, you go take care

Gets an F Z is equal to 20 x squared plus 20 y squared arrest by the focal point. Let's simplify this a bit so that his ex right over it won over 20. That's why I squared over one over 20. Okay, well, if we set, why couldn't you know? We get that the people, too went over 20 grown up. All right. He is equal to 20 x squared, and that's equal to X squared is equal to went over 20 Z, which is equal to four p. Z. Based on this, we know that p Well, we have four times something 1000 p. Three other dames, right? Uh, for me. Right. So we have won over 20. Difficult for PC. Based on this information we can solve for P, he is equal to won over 20 times. One over four peas. 1 80 More Focal point is this athletic culture zero common zero Mama P P is equal to won over 80


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