 In this video, we are going to complete some calculations for approximating the volume of a t-ondang lantern. If you're interested in the equations that we use here, you should see the accompanying video where we discuss the generation of those equations. So we assume over here that we have a series of data points that have been calculated or that have been measured along the sides of the t-ondang lantern, that we've measured up the side of the lantern when it was laid out flat and measured some heights, and we have some associated widths along with each of those pieces. Now, each of these are points that are at a particular place where we sliced, but what we're interested in is information for sections between any two of those points, which we're labeling with the same letter, but since we're doing something in a spreadsheet fashion, and by the way, if I was doing this calculation normally, I would often do it using a spreadsheet. So what we're gonna be looking at is the information between pairs of points. And we're gonna be writing that over here, so we're looking at sections that are in between segments. So let's start by finding out my change in h. How much does each section change in height along the sides of the section? And that math is fairly straightforward. You can see here that these were chosen to be increments of around 20 centimeters until we got near the peak. So in that particular case, I take each of these and I combine them and notice I'm gonna do my best to write them sort of halfway in between. Here's 20.0 centimeters for that first segment there. We're gonna label that with an i. Then we have 20.0 for the second and third and fourth segments. And then our last segment has a difference of only 10.5 centimeters. Notice again, if this was in a spreadsheet, we couldn't exactly shift it down that way. However, what I like to do in a spreadsheet is often mark it with a different color, something to indicate that the rows here correspond with some calculation across two rows in these columns. Second part of our calculation is going to be to calculate the change in width in both of these places. However, what we're really interested in, in this particular case, is not just the change in width, but we're gonna recognize that that change in width is split into two different locations and we really want the change in the width divided by two as it's on either side of the trapezoid and that's used in our calculation of figuring out this height of each frustrum. The frustrum being the shape of the section. So we're gonna write those values over here and I have pre-calculated those, so let me go ahead and write those values here. We have 7.7 centimeters, we have 4.9 centimeters, we have 0.1 centimeters, we have negative 8.9 centimeters and finally we have some staggering these in the middle there, so let me see if I can line those up just a little bit better. 0.1, negative 8.9 and finally a value of 19.3, negative 19.3. Now, the negatives might seem to be a little worrisome except we don't really have to care about them here because in a moment we're gonna square them because the formula we have for this height here takes these two values, our formula for the height uses the Pythagorean theorem and we're basically gonna take the square root of this delta H and square it and we're gonna subtract off the square root of this delta W over two squared. So since we've calculated each of those separately, we just need to go ahead and do the math there. We can put it into the spreadsheet, we can do it by hand however we'd like so I've already calculated some of those values here and I'm gonna go ahead and write them and it looks like I can get about three sig figs, maybe just two if we're looking at the numbers here but I'm gonna go ahead and record the three significant figures here so let me record those values. I get 18.5 centimeters there, centimeters there, 19.4, 20.0, the difference there is too small to really make a difference, 17.9 and then I see I have a little bit of a problem because if I look here, I see my delta H is only 10.5 and when I square that, well, I'm gonna get a bigger number somewhere around 100 then I look here and I see this value is 19.3 and when I square that, that's gonna be a bigger value and so I'm going to be subtracting a larger value from a smaller value giving me a negative value and then I have a square root of a negative value which is going to be an imaginary number. Uh-oh, what's the problem here? When I do that in my spreadsheet, my spreadsheet gives me an error and that error creates havoc all over the place. Well, what has happened here is our approximation is coming up with some problems. This last piece here at the very top of the lantern, what's happened is as we're folding it down, we don't have enough distance along H to reach and actually close the top of the lantern. Now in reality, the lantern was closed, it was touched in there, but my approximations, however they work, do not quite allow me to fill that in. So now I have to make a decision. Do I go back and change my approximation? Do I go back and measure more carefully? Or what exactly am I doing here? Well, one thing I am gonna recognize that if that's the case where it's flattened all the way out and can't quite reach, what is the volume of that piece that we're talking about? Effectively, we're going flat, we're trying to reach and we've completely flattened the top of the lantern. And so in that case, I have to make an executive decision but I'm realizing that even if I don't quite reach there, what volume am I looking for? Something that's perfectly flat has no height, no height? Well, let's just record that as zero. That's gonna give me a zero for my volume. So even though the real calculation gives me a negative imaginary number, I'm gonna estimate it as being zero. In a spreadsheet, I might put something like that saying if these values are, if the square of this value is bigger than that one, then record my value as zero and let the spreadsheet make that determination for me. Notice if that's substantial like this one kind of is, maybe I need to go back and rethink, maybe take more measurements so that near the top, so that perhaps I'm making a better estimate or maybe even rethink my approximation. In this case, I've done this a number of times and I believe even though this seems fairly substantial that it's not contributing a major change or difference into my overall volume. Okay, well now I have these height values. You'll notice there's the height values right there in my formula, my Heronian mean, this value here is my Heronian mean, which takes the height of my frustrum and then it multiplies by these values here, which are my widths, my width here squared, my width I is going to be one of my two widths, the bigger one. I square that, I also square the smaller one and then I multiply the two together and those three products I add together, divide by three and multiply by the height. Okay, that's a fair amount of work, but not very difficult on a spreadsheet. So I'm going to go ahead and record the values that I've calculated here. Let's see what they look like. I get actually first question, what are the units? Centimeters, there's a centimeter there and there's centimeters squared in each of these, so this is going to be in centimeters cubed or we could also record that as cc's cubic centimeters. So there's our volume there. Let me go ahead and record each piece. The volume for this piece is 28 and I'm going to write 00 and we'll go all the way down to the nearest cubic centimeter, although we probably do not, cannot quite justify that part. That I'll write that there now. So here 51, 184, I get 63, 393, I get 40,883 and then finally I get zero because the height is zero, it doesn't matter what the other values are. So now I have the volume for each of five different sections, effectively four sections. And so now what I do is I add up all those volumes to give me a final volume total for this example. My final volume total there is 183,471. However, this is again where I got to think about my significant figures at best I have, maybe two, we might consider this, well three significant figures in the data that I have here. So let me go ahead and write that as 183,000 and if I use my rule of treating a one as part of a significant figure, I think the best I could keep here 183,500 cubic centimeters as an approximation for the volume of this lantern. Could I do better? Two ways I can go about doing better. One way is I could think a little bit more about the geometry maybe instead of approximating these as a pyramid, the frustrum of a pyramid, I could do the frustrum of a cone and get the more circular nature of it. The other thing I could do is I could do more slices instead of only taking the six measurements that I have here, maybe I could take more measurements every 10 centimeters or every five centimeters, more accuracy, more work, but I could do that and then maybe have a little bit more faith in the accuracy and definitely get more precision for my estimate here. But this is probably a pretty good estimate and we'll go with that.