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A Pyramid is 680 ft high (due erosion: its currenr height is slighcly less) and has square base of side 4760 f. Find che work needed ro build the pvramid ifthe dens...

Question

A Pyramid is 680 ft high (due erosion: its currenr height is slighcly less) and has square base of side 4760 f. Find che work needed ro build the pvramid ifthe density of che stone estimated at 204 Iblfr '. 212778.[0 ff-Ib

A Pyramid is 680 ft high (due erosion: its currenr height is slighcly less) and has square base of side 4760 f. Find che work needed ro build the pvramid ifthe density of che stone estimated at 204 Iblfr '. 212778.[0 ff-Ib



Answers

Built around $2600 \mathrm{sCE},$ the Great Pyramid of Giza in Egypt (Figure 7) is 146 $\mathrm{m}$ high and has a square base of side 230 $\mathrm{m}$ . Find the work (against gravity) required to build the pyramid if the density of the stone is estimated at 2000 $\mathrm{kg} / \mathrm{m}^{3}$ .

My great walking back. So for this problem working yet? Well, pyramid here and all right, we've got these. I was right here. So this is actually, you know, this instance right here. 200 incriminating feet, Mr Centric. Here it is. 100 feet. Um, what makes this distance right here, Which is an important distance Quote. 100 feet. This single inside here. See what? 40 0.3 missing, right? Curious. He put a 46 27 degrees. All right, so knowing that begin, friend, is thing going here, it just can't be 90 minus 46.27 degrees. Just equal to 43.73 43.73 Not from this England Here. Right. Using 180 is 40.3. It is 43.37 So that will be que, uh, 95. Well, that doesn't seem quite right. I'm sorry. This is not, um, more on more. So I did that thing go wrong right there. 180 minus 46.27 So shot at this angle right here. Who's the single reading? Here's one to take over the 133 73 degrees. Sorry if I confused for a moment there. All right. Not would make that angle right there. Um, equal to 180. Minus 40. Went three. What is 1 33.73 degrees in that he wouldn't you? 5.97 degrees. That seems to fit the picture a little better. All right, so now here, Um, so for other one of the sides, right, and a decide or this side. Um Oh. So for, uh, this side great here should work. Okay, So, um, 100. My sign of 5.97 equal to this side will call this I decide. A provided by sign of 40.3. I could multiply both sides a sign of, uh, 40.3. And, uh, we will give us that A is equal to 641. Um, 28 7 feet. Great American friend. The height of this pyramid. So the height of the pyramid. Um, well, this single revolution before her 90 degrees, which makes this single right here the quarter of 180 when it's 90 minus 46.2. You're seven. That's when two equal to 43.73 degrees. And if it is 43.73 degrees, then UM, let's see inch using 12 signs. So eight divided by 46.27 reason is equal to that's, you know, outside is you quote you. It's a quarto 621.87 feet, divided by sine of 90 right? Of course, that is just one. So you get 6 21.7 sign 46.27 Well, that use us with H is equal to or 149 feet. That is the height of the period.

In this video, we will solve the triangle shown using love signs in order to find this length D. Which in this problem is the height of a pyramid. The information given is in green. First we can find the length C. Which will be equal to 200 -100 or 100 ft. Next we can find the angle be which is equal to 1 80 -D. Or 180 -46.27 Or 1 33 .73° next week. And find the angle see which will be equal to 1 80 minus a minus B. Or 5.97°.. And then we can use the law of sines to find this length B. You can plug in our information two yet That B is equal to 6 94.76 ft. Next we can use that sign of A. Is equal to D. Over B. Using this angle and the opposite and hypotheses. And so we know that D. Equals B. Times sign A. And we can plug in our information to get that D. Is equal to 4 49.36 ft, and this is our final answer.

So essentially the important information for this problem is that the sides of the pyramid base measure 5.6 inches in the pyramid is a 0.9 inches tall. We want to estimate the slope that a face of the pyramid makes with its base. So I have attempted to draw a visual here what we're trying to find so on the of the pyramid. So here's the the base early tried on this may be the base here of the pyramid is in total, um, 5.6 inches. All right, that's the length of one of the side one of the base, one side of the pyramid. And that means that if I just look at half the pyramid or half the length of the base of the pyramid, that will be 2.8. Great cause 2.8 toes 2.8 equals 5.6. And that'll be important, because now, so the because the hype here So it's a goes down to hear the height is what does it say? 8.9 interest. All. So this height here is a 0.9 inches, and I also know what the base is so let's and I want to find I want to find essentially. Like, what is the mom? What is this slope here? Right. So if we draw like a picture here, straw like this, this is kind of like, ah, side of it. Cut out. Okay, so this is 8.9. That's the hype. And then, um, half of the base years, 2.8, and again, we're looking at half of the base because, um, the height of the pyramid is in the middle. Right? That's like the tallest, the tallest part of the pyramid. The height is in the middle. We're looking at the middle of the bases. Well, but then we want to know. Okay. What is the slope? Well, Slope, you can think of this as changing. Why? Over change in X or rise over. Run. And here the rise is 8.9, and the run is 2.8. So plug it in on your calculator. 8.9, divided by 2.8. And we get about We'll do this about three point two, um, inches. And that is the slow

Okay, we have an inverted pyramid with a square base um it is two by two m at the top and it has a foot height of five m. So we're gonna pump water Well actually um slurry over the side. So we're given a density of that corn slurry. Okay, so to start off um we do have a depth of four m. So we are going to be going from 0-4. They've given us the density of 17.9 and then it's not a weight density, it is just a mass density. So we're going to have to multiply by 9.8. The next thing is, I'm putting in my five minus X because notice the depth is only four m. That's why we're going from 0 to 4, but the height is up at five. So if you think about something at the very, very bottom at zero, if you placed in it would have to go five up and then every number is going to be placed in until at the last will be placing in force and they will only have to be pumped um one m. And so you can see how those two relate. Now the last thing I need is my volume, which is gonna be my cross sectional area, multiplied by dX because the dx is the is going to be the thickness of that cross sectional area. So to come up with that cross sectional area, I need to consider, you know, the I'm going from like zero at the bottom when I'm going to have a zero times zero, but then when I'm at the top I'm going to have a two times two. So I'm thinking how do I put in a zero to get out of zero? But then put in a five to get out at two? Well that really has a slope of 2/5. So now anything that I want to put in between them will um end up giving me kind of proportion to that value. So that value goes in as my radius or my side and then I have to square it because it is a square. Okay so now I can move some of my numbers to the front but then I'm going to have to distribute that for over 25 X squared into five minus X. And so I end up with a four or five X squared minus A. For over 25 X to the third. Okay so now we're finally ready to integrate and so get those constants um but then I can go up a power to three and divide by three. So all the 4/15. Then I go up to the power of four and divide by four. So now I'll just have a 1/25 I'll be placing in my four and then subtracting putting in zero. But of course when I tracked putting in zero that is zeros out so I can consider putting my for ins um end up with a four to the fourth because the four and then the four to the third and then the other one. I also have a four of the fourth. Um And so really when I throw it in my calculator, I end up just putting that four the fourth and then I multiply by that parentheses of 1/15 minus one 25th. But however you want to do that um You should come out with a answer of 1197.5 jewels.


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