Applying volume of solids

- [Instructor] We're told that a cone-shaped grain hopper, and they highlight hopper in blue here, in case you want to know its definition on the exercise. It's something that would store grain, and then it can kind of fall out the bottom, has a radius of 10 meters at the top and is 8 meters tall. So let's draw that. So it's cone-shaped, and it has a radius at the top, so the top must be where the base is. I guess one would think about it. It must be the wider part of the cone. So it looks like this, something like that. That's what this first sentence tells us. It has a radius of 10 meters, so this distance right over here is 10 meters, and the height is 8 meters. They say it's 8 meters tall, so this right over here is 8 meters. Then they tell us it is filled up to 2 meters from the top with grain. So one way to think about it, it's filled about this high with grain, so it's filled about that high with grain, so this distance is going to be 8 minus these 2, so this is going to be 6 meters high. That's what that second sentence tells us. The hopper will pour the grain into boxes with dimensions of 0.5 meters by 0.5 meters by 0.4 meters. The hopper pours grain at a rate of 8 cubic meters per minute. So, a lot of information there. The first question is what is the volume of grain in the hopper? Before we even get to these other questions, let's see if we can answer that. So that's going to be this volume right over here, of the red part, the cone made up of the grain. Pause this video and try to figure it out. Well, from previous videos, we know that volume of a cone is going to be 1/3 times the area of the base, times the height. Now we know the height is 6 meters, but what we need to do is figure out the area of the base. Well, how do we do that? Well, we'd have to figure out the radius of the base. Let's call that r, right over here. And how do we figure that out? Well, we can look at these two triangles that you can see on my screen, and realize that they are similar triangles. This line is parallel to that line. This is a right angle. This is a right angle, because both of these cuts of these surfaces are going to be parallel to the ground, and then this angle's going to be converted to this angle, 'cause you could view this line as a transversal between parallel lines, and these are corresponding angles. And then both triangles share this. So you have angle, angle, angle. These are similar triangles, and so we can set up a proportion here. We can say the ratio between r and 10 meters, the ratio of r to 10, is equal to the ratio of 6 to 8. Is equal to the ratio of 6 to 8, and then we could try to solve for r. R is going to be equal to, r is equal to... Multiply both sides by 10. Multiply both sides by 10, and you're going to get 60 over 8. 60 over 8, 8 goes into 60 seven times with 4 left over. So it's 7 4/8, or it's also 7.5, and so if you want to know the area of the base right over here, if you wanted to know this b, it would be pi times the radius squared, so b in this case, is going to be pi times 7.5, we're dealing with meters squared, and so the volume, to answer the first question, the volume's going to be 1/3 times the area of the base, this area up here, which is pi times 7.5 meters squared, times the height, so times 6 meters. And let's see, we could simplify this a little bit. 6 divided by 3, or 6 times 1/3, is just going to be equal to 2, and so let me get my calculator. They say round to nearest tenth of a cubic meter. So we have 7.5 squared times 2, times pi, is equal to, well if we round to the nearest tenth, it's gonna be 353.4 cubic meters. So the volume is approximately 353.4 cubic meters. So that's the answer to the first part right over there, and then they say how many complete boxes will the grain fill? Well, they talk about the boxes right over here. The hopper will pour the grain into boxes with dimensions of 0.5 meters by 0.5 meters by 0.4 meters, so we can imagine these boxes. They look like this, and they are 0.5 meters by 0.5 meters by 0.4 meters, so the volume of each box is just going to be the product of these three numbers. So the volume of each box is going to be the width times the depth times the height, so 0.5 meters times 0.5 meters times 0.4 meters, and we should be able to do this in our head, because 5 times 5 is 25. 25 times 4 is equal to 100, but then we have to think we have one, two, three digits to the right of the decimal point, so one, two, three. So this is going to be a tenth, 0.100 cubic meters, so a tenth of a cubic meter. So how many tenths of cubic meters can I fill up with this much grain? Well, it's just going to be this number divided by a tenth. Well, if you divide by a tenth, that's the same thing as multiplying by 10, and so if you multiply this by 10, you're going to get 3,534 boxes. Now once again, let's just appreciate. Every cubic meter, you can fill 10 of these boxes, and this is how many cubic meters we have, so if you multiply this by 10, it tells you how many boxes you fill up, and one way to think about it, we've seen this in other videos, we're shifting the decimal one place over to the right to get this many boxes, and, it's important to realize, complete boxes, because when we got to 353.4, we did round down, so we do have that amount, but we're not going to fill up another box with whatever this rounding error, that we rounded down from. So the last question is, to the nearest minute, how long does it take to fill the boxes? Well, this is the total volume, and we're going to fill 8 cubic meters per minute, so the answer over here is going to be our total volume, it's going to be 353.4 cubic meters, and we're going to divide that by our rate, 8 cubic meters per minute, and that is going to give us 353.4 divided by 8, is equal to, and if we want to round to the nearest minute, 44 minutes. Is equal to approximately 44 minutes to fill all of the boxes, and we're done.