ABSCHLUSSPRÜFUNG Realschule Mathe – Geometrie 10. Klasse

Hello dears, today I bring you a task from a high school final exam, this task occurred in Bavaria in 2022 and it goes like this: the template for a candlestick for spherical candles is a body of revolution with the axis of rotation mn It is already starting, let's take a quick look at the sketch what this is, the red one drawn here is this candlestick that was supposed to be created for spherical candles, so this spherical candle is supposed to be put here. So the red is this support, everything else that is drawn in dashed lines are just auxiliary lines.

You have to imagine the axis of rotation mn here from ma N. There is also an arrow drawn here so that everything drawn here rotates. this axis, so it rotates all the time, so to speak, and the body that is then created around it, is it possible that it is this chandelier? The sketch shows the axial section of this body of revolution exactly once here on the axis of rotation. As we just described, point C is the intersection of the straight line from and ED, so if we draw a straight line through A B here and so extend and also extend through Ed and here, then the two straight lines intersect here at point C, which is how it all looks, then we get longitude information that we won't look at in detail now, maybe just here that ae runs parallel to BD.

This little part here runs parallel to this one, just as you would like. to be and as you imagine it, that should definitely be the case. Part A calculates the volume of the body of revolution, rounds to two decimal places, okay, so we have to make rounding errors. Be careful, there would be a point deduction. , we should determine the volume of this candle, maybe we can look at part A first and then part B. So let's go to a new page first, how can we now determine the volume of this body of revolution? That is if. volume of this red part if it rotates around this axis over and over again, maybe first we give it a name that we say is the volume of this rotating body, so ultimately VR is what we are looking for, but we don't have a Formula that we can apply to it because virtual reality seems wild.

No, it is not a classic figure. It is not a cone, it is not a cylinder, it is not a ball, it is simply made up of other figures that, thank God, are already drawn. No, we can already see figures around him that look familiar to us. For example, if we draw this big triangle here that encompasses everything but it would be too big, but you can start with that if we determine the volume. of it, of course not of the triangle itself but also of the one that rotates around its axis, what is created when a triangle if you let it rotate, what type of body is created?

You can imagine this turning into a circle in this area of ​​the base. here and here towards the tip it tapers like this and that's a cone, so if you import a triangle like this you just create a cone and of course we can calculate the volume from that. Then we have to do V1, but that's too much. big for what we really want. We still have to subtract some of the red body. Well, this piece here, this point is definitely too much, this is a triangle again, it will become a cone again, let's call it V2, the volume. from this cone, and then we also have to subtract something else, namely this part here.

In the middle I will make it blue, what is it because what will it be if you look at it as a volume behind three times and subtract it? here if you let a semicircle rotate like this, what is created? You can see that here it also becomes circular and that it simply becomes a hemisphere because this candle also enters, this spherical candle and that it is like this inside spheres so let's also watch the volume, okay, so we have a plan for what we're going to do now. I would say let's calculate V1, V2 and V3 individually on individual pages because you can see it's getting extensive so we have a bit of work to do.

I hope it doesn't get too tangled but let's go in first and find the volume of V1. We have already said that V1 will become a cone, which means that we would first need the formula for the volume of a cone. The cone has radius r and we call the height that goes from the center to the tip. Then you can calculate the volume as follows. The formula says one third times the area of ​​the base, which here is a circle, so the area of ​​the circle is pi times R squared and everything is simply multiplied by the height.

This is our formula that we are now using in our case, which means we are using one third We can take PI, what does it look like with R? ?In our case, the radius of our green cone will be this piece here, from m to E and taking it first, so the distance mE is our radius, which still remains to square the formula and multiply everything by the height, which It goes from the center to the top, so first enter the distance from C to M and then you will get points. It hasn't been miscalculated yet.

Somehow it's all there. That's all. It's pretty good, so we can see. If we have given information about the length, how about me this piece here? It's given somewhere no, it doesn't look good, the distance is given here ah, it's okay with 9 cm, but we just want to have this piece, the meditation axis is of course. in the center of the body of revolution, which means that this piece is simply half of the entire distance, so half of 9 is then the 4.5 cm that we can enter here and in my opinion enter one into the formula. I take the units now. add the centimeters to the equation, we have to see if we have to do it or not, do it properly or leave it out as it has to be done, but then we have to put a bracket around it, square everything because that is actually just this section of 4.5 and behind him, what about our cm? cm is this piece here which is not given CN is given with 5.5 this is the short piece here mn unfortunately it is not given BD we can still see that piece here from there to there if you just take half of the 5 cm the short piece now It's 2.5 long, yes, but we couldn't find that piece.

You definitely have to calculate it. You'll notice something on this right side when you look. Let's see what kind of figure it is here, we have here the piece that runs parallel to it, then we have two straight lines that intersect at a point here in the so-called center, you will notice that this is a ray. set the figure and we can do that with that. So with the ray theorem, calculate here this desired distance in cm that we need for our formula, but someone can do it on a new page and maybe first let's draw it a little bit cleaner so we can see a nice figure of the ray theorem. ray down here is our C here is our m the side here is 4.5 long and 2.5 long and 5.5 these are the things that we have given, we want to calculate that, but given the short blue path and the ones that runs parallel here I mark the long one sometimes in green, the short one is also green, these are the two parallels and we can use these four segments in the ray theorem, which says that the long blue segment is related to the long green parallel segment of the same way the short blue segment relates to the short parallel segment, so write them down.

The distance relates to the long parallel piece, i.e. 0.5 cm, in the same way as the short blue piece, 5.5 cm, to the short green parallel part, i.e. 2.5 cm. . This is simply an equation that we can rearrange according to our desired cm. When calculating 4.5 cm here we can also eliminate centimeters and centimeters and then find our desired cm by dividing the 5.5 by the 2.5. and multiplying everything by the 4.5 cm that can be numbers. Just plug it into a calculator and you'll get a distance of 9.9 cm for our cm, which is what we needed for this formula here, then we can plug it in here and We've found our first volume.

Now we can do a little bit here. I wouldn't type it into a calculator and calculate it thinking about the rounding errors that can occur, so let's be exact, maybe loosen the parentheses a little here. little to 4.5 squared and we don't have the square centimeters between here and back here still the new comma 9 cm in the back if now we sort a little so that we have one unit here and one unit here, everything else that can be written in the calculators, let's introduce it and what unit we have in total square centimeters multiplied by centimeters, then to cubic centimeters as is appropriate for a volume and then we would leave it as it is, we would not write it in the calculator, we know that it is calculator ready, but we want everything to be exact so there are no rounding errors, so do it like this, let us know, but we can check V1, we can access it at any time and then we will take care of V2 but that will now be much faster because it is another cone and we are already trained for the volume of a cone one third of Pi R multiplied by the height, we use our second volume for this, that is, one third of Pi, which is the radius, thank God, these are given here , all with a nice 2.5 cm support around and square everything and the height is also given here at 5.5 cm, so we don't need to use another set of spokes. which means that if we embellish it a little we have what the parentheses we can solve here: 2.5 squared, the centimeters also have to be squared but then directly with the centimeters back here, centimeters at the height of 2 times cm are centimeters at the height of 3 and we cannot forget the 5.5, we add the delicate ones to the numbers, that is our second volume, we did not calculate that either, we just remember that we would have it and then we get to the third volume. , which is a little bit different now because it's about this hemisphere that is created here, so in this third volume we have to see what the formula is if we first look at a sphere, there is a formula for the volume of a sphere that says four the third time times Pi times R to the power of 3.

Then you just need the radius of the sphere and you can calculate the volume. Now we have to be careful that we don't have a sphere it's a hemisphere We have here so we only need half so we don't need a sphere but a hemisphere and then we can put one and a half in front because we really just want to have the half the volume of a full sphere, okay, so add half a quarter third of everything multiplied by Pi, okay Radius, all we need is what the radius looks like here that's drawn, that would be this piece here is MF since GF is given, so depending on F is 5 cm and here it is Also since the radius has the same length as mg and MF, then half of the five, that is, two point five centimeters, we can put it here and then not forget everything according to the formula raised to 3. and then we can do it a little better.

I wouldn't have many. Think about reducing some fractions or something, but just copying them. We'll write a calculator anyway, but back here we'll do the same thing. 2.5 to the power of 3 and cubic centimeters so that we have our third volume Guys, we have given V1 V2 V3 exactly We can enter this in our formula Enter it in the calculator Round to two decimal places and get a volume of 141.21 cubic centimeters We have done the ah part, let's see What else does B have in store for us? The candelabra must be made of marble. One cubic centimeter of the used marble has a mass of 2.7 grams.

Calculate the mass of the round candlestick to whole grams. Well, we have the volume of our chandelier that we can use again because it is now given that one cubic centimeter of marble has a mass of 2.7 grams. But we don't just have one cubic centimeter, we have 141.21 cubic centimeters and then we can just calculate how many grams that corresponds to. It's basically just three sentences, the numbers are very nice. We can simply calculate 141.21 on the left side and do the same on the other side, 141.21. If we do that, First we get to 381.267 grams. We should round up to whole grams.

That's the last thing, so we'll round up to whole grams. That means that we will see them in the number that says round down, so we stay at 381 grams and with that we are complete. homework. Don't hesitate to write in the comments what you thought of the task. I await your evaluation. Have a wonderful day and see you next time. The video does it well.