 Hi, I'm Zor. Welcome to Unizor Education. I would like to continue solving problems about cones. As usual, I suggest you to go to unizor.com to notes for this lecture. This is the second problems lecture in the topic called cones. And try to solve these problems yourself. They are absolutely not difficult at all. And it will definitely be much more useful for you if you will do it yourself. Now, even if you will not succeed, and there are answers which you can check your results with just to make sure you are right, even if you are not succeeding, it's still very useful just to think about these problems, about how you would like them to approach, etc. So then, I will present my solutions. And after I finish the lecture, I would suggest you to repeat this process just by yourself again. Try to solve them using whatever information you have learned from the lecture. Alright, so let's solve the problems. Problem number one. Okay, this is a problem for my childhood to tell you the truth. When I was learning solid geometry, it was presented in one of the textbooks. I don't even remember where and how. So here is the problem. A little girl got sick, and doctor prescribed her liquid medicine, and he said to take from 15 to 20 grams of this liquid medicine at a time. Whatever number of times a day. Now, mom has this conical glass, which holds 20 grams exactly of the liquid. So, she poured it completely to the brim with the medicine, but the girl said, well, I don't like the taste of it, I don't want to drink it, etc. So, mom said, you know what, you're sick, you have to take medicine, but let's make a compromise. You can leave half the height of this glass. So, just drink half of it. Half means half by height. And, well, little girl agreed. Now the question is, how much of the medicine this girl really took if she left half of it by height? Well, let's just think about it. It's really very simple. If you have a cone, this is the radius, and this is half the height. Obviously, this radius is half, as well as this height is half of the height. So, this little cone, which contains the liquid which is left, has half the radius and half the height, but if you remember, the volume is this one, right? One third of the area of the base of this cone times height. Now, r is half, now square would be one quarter, h is also half, so it would be one eighth as a result, right? Because this is twice as small, and this is twice as small, this is square, so it would be one eighth of the initial volume. So, if the initial volume was 20, then this little thing would be 20 divided by 8, which is 2.5. Which means whatever is left is 20 minus 2.5, which is 17.5, right in the middle of this prescription. So, we are okay, and the girl will feel much better after she took only half the dosage. And mom was absolutely right to go for this compromise. Next, okay, next. If you have a cone, and as we did many times, if we cut it along one of the generatrix and roll it out, we will have a sector. Now, this sector will have the radius equal to length of the generatrix, which is L. And L, by the way, is square root of h square plus r square, but it's a Gorian theorem, right? And the arc would be equal in length to a circumference, which is 2 pi r. Now, the problem, if you have taken initially this sector, and it's not this angle, it's 180 degree sector of the radius, let's say, r. Let's not use this r here. And roll it into a cone. So, the question is, what would be the volume of this cone? Well, let's just think about it. I'll use different letters now. If this is lowercase r, lowercase h, now this radius of this rolled out cone would be equal to square root of h square plus r square, right? So, we know that. Now, what else do we know? We know that the circumference is equal to the arc length, right? Which means that 2 pi lowercase r, which is circumference, equals to half of this, which is 2 pi r divided by 2, which is pi r. So, we have two equations with two variables, lowercase h and r. And we can obviously solve this, find it, and then, using the formula for the volume of the cone, get the volume, right? So, how to solve it? It's very simple. This, obviously, is reducible by pi. So, lowercase r is half of the uppercase r. Now, we are substituting into this. So, we have already found lowercase r, right? Now, now h. If you will substitute r here, you will end, let's square both of them. So, it's r square is equal to h square plus lowercase r square, which is r square divided by 4. So, h square is equal to 3 quarters of r square, and h is equal to r square root of 3 divided by 2. So, we have h. We have h and we have r. So, the volume is equal to 1 third pi r square h equals to 1 third pi r square, which is r square by 4, and h is r square root of 3 by 2. Equals 2 pi r cube square root of 3 divided by 24. So, that's the volume of the cone, which is made of this half a circle of the radius capital R. Okay, next. Next is... Okay, now imagine you have a cylinder inside a cone. Here is how it will look. So, this is my cone. Cylinder has a base, which is standing on the base of the cylinder inside. So, it would be something like this. And it goes up. That's my cylinder inside. Now, what's known about this? Now, obviously, the axis of the cylinder and cone coincide. Now, we know the radius of the cylinder is lowercase r, radius of the cone is capital R. And I know the height of the cone, h. So, my problem is to find the volume of the cylinder. Alright, that's actually a very easy thing. Consider these two triangles. Well, let's say we have a plane which goes through the main axis. Now, what would be in the section of this plane? Well, obviously, you will have this from the cone, right? And this from the cylinder on the plane. That's what we'll do, right? So, this is the apex. Now, when the plane cuts it, that would be a diameter of both of them, actually. This is a diameter of a cylinder base and this is a diameter of the cone base. And I know basically everything. This is r, this is capital R, this is h. So, all I need right now is to know the height of the cylinder, right? If I know this h, I will know the radius and the height, so I will know the value. Now, here it's a very simple thing because you have obviously similarity of this small triangle and the bigger triangle. Now, in the small triangle, one calculus is h and another calculus is capital R minus lowercase r. So, this piece, r minus r. So, the ratio between these calculus is equal to the ratio between these calculus, which is h and r. Now, that's sufficient to find h and then the volume, right? So, that's the plan. h equals to hr minus r divided by r, right? From this proportion. And the volume is equal to pi lowercase r square times h, which is hr minus r divided by r. That's the volume. Yes. Next. Okay, for instance, you have, we did it actually many times, but not in this particular context. So, you have a cone, radius r and height h. Now, you cut it along the generatrix and roll it out, so you'll get a sector. So, my question is, what would be this angle? Let's say in regions. Okay, let's think about what this sector actually represents. Well, what I know is the radius of the circle this sector is a part of, right? This is the generatrix, which is l equals to square root of r square plus h square. This is l. This is l. Now, what is the arc? The arc length is the circumference. This is 2 pi arc. So, I know about this sector, the radius of the circle it's part of, and I know the arc lengths. Well, obviously I can find out the angle, because the angle is proportional to the arc lengths, right? The full angle 360 degree, which is 2 pi radians has 2 pi l lengths of the whole circumference, right? Full angle, so it's the whole circumference, so it's 2 pi square root of r square plus h square. Now, my angle phi in radians, this is radians and this is radians, is measured by this arc. Or arc is measured by angle. So, if arc is this, then the angle is this. If arc is this, the angle I can find out, right? So, what would be the angle? Well, 2 pi obviously goes out and 2 pi r goes here. So, phi is equal to 2 pi r divided by square root of r square plus h square. That's the answer. So, whenever I have a cone, I have characteristics of this cone, radius and altitude, the fight, altitude. Then I can find out the angle, which is the result of my rolling out this cone on a flat surface. This would be the angle in radians. Okay, and the last problem. You have a cylinder and inscribed into this cylinder. There are two cones. One cone has exactly the same base as the cylinder and the apex coincides with the center of the upper base of the cylinder. So, this is the cone I'm talking about. Another cone, exactly the same as this one, by the way, has the base on the top and it goes down, the apex is here on the bottom. And this is the intersection. So, the intersection between these cones is a geometrical object, which is kind of inside of both cones. And what is this? Well, it's actually, if you think about it, if you have these two cones like putting, I don't know, inserted into each other. Kind of inserted into each other with their heads, with their apexes. So, it's a cone which is on the top and then you have a cone which is on the bottom. That's what it probably would be. So, this kind of complex geometric object is actually a combination of two cones. Because obviously, if you cut the whole thing in the middle, you will have this particular section, which is boundary between these two cones. So, what I have to do is, well, I don't know, did I state the problem? The problem is what's the volume of this common part of the intersection between these two cones relative to the volume of the whole cylinder. So, let's just evaluate one half of this figure, which is one cone. And that's very easy because it's right in the middle, which means it has half the diameter, which means half the radius. And it has half the height, right? So, if my initial parameters of the cylinder are, let's say, r and h. So, the volume is pi r squared times h, right? Now, what would be the volume of this cone? Well, the volume of this cone would be half the radius and half the... So, it's one-third times pi r divided by two-half squared and h divided by two times, right? That's volume of this lower cone and same thing with the upper cone. So, combined, that would be times two. So, that would be the result. This is the volume of the common part, which is equal to, well, volume of the initial cylinder v divided by how much? This is four and this is two, which is eight times three-twenty-four and this is two. So, it's one-twelfth, right? Am I right? So, it's one-twelfth of the cylinder. Or, I don't know, if you want which part of each cone it will be... It's obviously one-third of that, so it will be one-fourth. So, relative to the cylinder, it will be one-twelfth. Relative to the cone, one of these two cones, it will be one-fourth, because I will not have this one-third, right? So, that's the answer. That's it. I do suggest you to go to the notes on unizor.com for this lecture and try to solve all these problems yourself, accurately on the paper and check against the answers which are presented on the notes for this lecture. Well, that's it. Thank you very much and good luck.