Estimating volume of hemispheres

0:00- [Sal] A hemisphere with radius eight centimeters fits snugly inside a cylinder of the same radius and height. We can see that here. What is the best estimate for the volume of the hemisphere? So, pause this video and have a go at that. Okay so, the way I'm going to estimate this is I'm going to actually find an upper bound and a lower bound.

0:18So, the upper bound is going to be the volume of the cylinder that this hemisphere clearly fits in. So this hemisphere and the cylinder touch right over there, eight centimeters is also the height of that cylinder. So whatever the volume of the cylinder, which we know how to find, that's going to be an upper bound for the volume of the hemisphere.

0:37So how could we find a lower bound? We know it's something that the hemisphere has to be larger than. Well, we can imagine a cone that's inside of the hemisphere. So, let's imagine a cone that looks like this. If we figure out its volume, that will fit inside the hemisphere, so its volume is going to be less than the hemisphere.

1:00So let's first do, let's first do, let me see where I can have some space, the volume of the cylinder, cylinder, is going to be equal to the area of the base, which is pi times the radius squared. The radius is eight centimeters, so that squared is 64, so it's going to be 64 pi times the height times eight.

1:27Times 8, so this is going to be equal to 512 pi. And then let's see, the volume, I'll do it down over here, the volume, I should do in that pink since I drew the cone in pink, So the volume of the cone is going to be equal to 1/3. It's actually essentially 1/3 times the volume of this cylinder right over here.

1:51So it's going to be 1/3, because 1/3 times the base, the area of the base, times the height. The area of the base times the height is the volume of the cylinder, so it's 1/3 the volume of the cylinder, so it's 1/3 times 512 pi. So that's equal to, let's see, this actually is not straight up divisible by three, but I'll just write 512 over 3, or 512 pi over three.

2:15So, the volume of our cone is between these two, or sorry, the volume of our hemisphere, is between the cone and the cylinder. So, let me write this in another color. So the volume, the volume of the hemisphere, hemisphere is going to be greater than, actually it's going to be strictly greater than 512, 512 pi over 3, and it's going to be less than 512 pi.

2:47So, let's think about this. They don't give it in terms of pi here, so it's going to be, this is going to be over, if we approximated pi as three, then, actually why don't I just get a calculator out just to get a better estimate of this? Well, no, no, they're telling us to estimate, so I don't want to use a calculator.

3:08All right. (chuckles) So, if I were to say this is going to be, this right over here, is approximately equal to, if I just multiply it by three, that'd be 1,536. But if I multiply by 3.14, you're going to add another, let's see, if you just multiply by 3.1, you're gonna add another 51, so it's going to be approximately 1,600 over here.

3:28And this one is going to be approximately 1,600 divided by, divided by three, and so that's going to be, well, give or take, I don't know, approximately 500 or so. And so we actually have two reasonably good candidates here. We could either have the 1,070 cubic centimeters or the 800.

3:52Now, the key is, for it to be 800, this hemisphere has to be less than half of the cylinder. And this definitely does not look like less than half of the cylinder, obviously, we are just estimating here, so that's why I'm going with that, and so that's why I like the 1,070 cubic centimeters.

4:11In future videos, we'll actually learn the formula for the volume of a sphere. I could tell it to you, but anyway, you could look it up, and then obviously you would take half of that for the hemisphere. But we will do that, we will do that at another time.