Applying volume of solids | Solid geometry | High school geometry | Khan Academy

We're told that a cone-shaped grain hopper, and they put the 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 of the bottom.

It 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 way to think about 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 eight meters tall, so this right over here is eight meters.

Then they tell us it is filled up to two 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 eight minus these two, so this is going to be six 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 the first—quite a lot of information there. The first question is, what is the volume of grain in the hopper? So 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 the 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 us call, 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.

Then this angle is going to be congruent to this angle because you could view this line as a transversal between parallel lines, and these are corresponding angles. 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, which is equal to the ratio of 6 to 8.

Then we could try to solve for r. r is going to be equal to—r is equal to—multiply both sides by 10, so divide 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 seven and four-eighths, or it's also 7.5.

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.

So the volume, to answer the first question, the volume is 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 six meters. Let's see, we could simplify this a little bit: six divided by three, or six times one-third, is just going to be equal to two.

Let me get my calculator. They say round to the nearest tenth of a cubic meter, so we have 7.5 squared times 2 times pi is equal to—if we round to the nearest tenth, it's going to be 353.4 cubic meters. So volume is approximately 353.4 cubic meters, so that's the answer to the first part right over there.

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. We should be able to do this in our head, because 5 times 5 is 25, and 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. 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 that for 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. 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.

If we want to round to the nearest minute, 44 minutes is equal to approximately 44 minutes to fill all the boxes, and we're done.