ABSCHLUSSPRÜFUNG Realschule Mathe – Geometrie 10. Klasse

Hello dears, today I brought you an assignment from a final exam in high school. This task was carried out in Bavaria in 2022 and goes like this: The template for a candlestick for spherical candles is a body of revolution with the axis of rotation mn. It's already starting, let's take a quick look at the sketch of what it 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 information about the length 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, just as you would like. be and as you imagine, 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 simply 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?

Can you imagine this turning into a circle in this area of ​​the base? here and here towards the tip it tapers like this and it'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 too. big for what we really want we still have to subtract some of the red body Well, this piece here, this tip 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? Do you 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, we will do it like this? 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's not too complicated, 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. It has the radius r and we call the height that goes from the center to the tip. Then you can calculate the volume in the following way. The formula says one third times the area of ​​the base, which here is a circle, so the area of ​​the circle. It's pi times R squared and everything is simply multiplied by the height.

This is our formula that we are using now 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 bit here, from m to E and taking it first, so the distance mE is our radius, which still needs to be squared with the formula and everything is multiplied by the height . , which goes from the middle to the top, so first enter the distance from C to M and then you get points. It hasn't been miscalculated yet.

Somehow it's all there. 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 fine 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, it which means that this piece is simply half the total distance, so half of 9 is the 4.5 cm that we can enter here and in my opinion enter one into the formula. I take the units.

Now I add. the centimeters in the equation, you have to see if you have to do it or not, do it accordingly or leave it out like you have to do it, but then we have to put a bracket around it, square everything up because that's really it's just this stretch of 4.5 and behind of him, what happens to our cm? cm is this piece here which is not given nor 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 can't find that piece.

You definitely have to calculate it. You'll notice something on this right side when you look at it. Let's see what type of figure it is here, we have the piece here that runs parallel to it, then we have two straight lines that intersect at a point here at the so-called center, you will notice that this is a set of rays. figure and we can do that with that So with the ray theorem, so calculate here this desired distance cm that we need for our formula, but someone can do that 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 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 run parallel here I mark the long one sometimes in green, the short one is also green, these are the two parallel and we can use these four segments in the ray theorem, which says that the long blue segment is related with the long green parallel segment in the same way that the short blue segment relates to the short parallel segment, so write them down.

The distance relates to the long parallel piece, i.e. the 0.5 cm, in the same way that the short blue piece, the 5.5 cm, relates to the short piece. parallel green part, that is, the 2.5 cm. This is simply an equation that we can rearrange based on our desired cm. Calculating 4.5 cm here we can eliminate centimeters and centimeters and then find our desired cm by dividing the 5.5 by 2.5 and multiplying. all for the 4.5 cm which can be numbers Just enter it into a calculator and you will get a distance of 9.9 cm for our cm which is what we needed for this formula here, then we can enter it here and we have found our first volume now We can do a little here.

I wouldn't type it into a calculator and calculate it thinking about the rounding errors that may occur, so let's be exact, maybe loosen the brackets a little here. 4.5 squared and we don't have the square centimeters between here and back here, the new comma of 9 cm still appears in the back if we now sort a little so that we have one unit here and one unit here, everything else in the that you can write calculators let's go ahead and what unit we have in total square centimeters times centimeters then to cubic centimeters as is appropriate for a volume and then we would leave it as it is, we would not enter it into the calculator, we know that it is ready for the calculator, 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 squared times the height, we use our second volume for this, that is, one third of Pi, what is the radius? , thank goodness these are given here, all with a nice 2.5cm support around and squares everything and the height is also given here at 5.5cm, so we don't need to use another set of spokes, which It 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 about 5.5, we add the delicate ones to the numbers, that is our second volume, we did not calculate that either, we just remember what it would be like to have it and then we get to the third volume, which is a little different now because it is 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 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 actually only want to have half the volume of a full sphere okay so add the half. fourth third of everything multiplied by Pi, okay Radius, all we need is what the radius looks like here that is drawn, which would be this piece here is MF since GF is given, so depending on F is 5 cm and here it is also given that the radius has the same length as mg and MF, so 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 bit better.

I wouldn't have too many. Think about reducing some of the fractions or something, just write them down, we write on a calculator anyway, but here we do 2.5 to the power of. 3 and cubic centimeters so that we have our third volume, people, we have given exactly V1 V2 V3 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 part ah, let's see what else we has reserved B. The chandelier must be made of marble, one cubic centimeter of the marble used 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 we can use it again because now it is 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 simply 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, we first 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 the number that says round down to stay at 381 grams and with that we have completed the task. 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.