Volume word problems — Basic example

- [Instructor] A tea infuser in the shape of a right rectangular pyramid is 7.9 centimeters tall and has a base three centimeters long and 1.5 centimeters wide. To make the best tea, the infuser should be 80% filled with tea. What is the volume of tea, in cubic centimeters, needed to fill the infuser to 80% of its capacity. Round to the nearest tenth. All right, so this tea infuser, it's kind of one of those fancy teabags that hold their shape, is in the shape of a right rectangular pyramid. So it's gonna look something like this. Let's see, its base is three centimeters by 1.5 centimeters, so it's base is, let's say that's three centimeters by 1.5 centimeters, so it might look something like that. So that's its base. Let me draw that. Actually, I can even label it, so this is three centimeters, this is 1.5 centimeters, and then it is 7.9 centimeters tall. So if we went from the center, and if we were to go straight up, it's 7.9 centimeters tall. So this dimension right over here is 7.9, 7.9 centimeters, and it's a right rectangular period, so you could kind of think of the pyramids in Egypt, although this one, this one's, this one's a little taller, relative to its base than those, but you might have seen these fancy tea infusers. So it's gonna look something like this, and if it was transparent, wou could be able to see this one right back here. So what we can do is, we can first find the volume of this right rectangular pyramid, and then they say that it should be 80% filled with tea, so we need to figure out what 80% of its volume is, and then that's gonna tell us what's the volume of tea needed to fill the infuser to 80% of its capacity. Now, you might be saying, how do I figure out the volume of a right rectangular pyramid? Well I'm about to tell you that. And there's a formula, I'm not gonna prove it here, especially if you're the middle of the SATs, not a lotta time for proofs, but the formula here, in some ways, is strangely intuitive. You multiply essentially the base, the area of the base times the height and then divide that by three. And so what's the area of the base, well it's going to be, it's going to be the length times the width, the length times the width. And then you multiply that times the height, so it's gonna be the length times the width, times the height, and then divide it by three. So in this case it's going to be, and just another way of thinking about it, this right over here, the length times the width, that's the area of this base, and then you multiply it times the height, and then you divide by three. If you didn't divide by three, you would get the volume of the cube, not the cube, I should say, the rectangular prism, it's not a cube, all the dimensions aren't the same. That would, that would contain this thing. But we're not concerned about the volume of the rectangular prism, we are concerned with the volume of the right rectangular pyramid. And so this is going to be, length, three centimeters, times width of 1.5 centimeters, times height of 7.9 centimeters, divided by, divided by three. Well, this three, three divided by three is just one. And let's see, you have centimeters, whoops. This is, actually, let me just change. This is centimeters right over here, divided by three. That cancels with that, you have centimeters times centimeters times centimeters, it's going to be centimeters cubed. So your volume's going to be, let's see, it's going to be 1.5 times 7.9 centimeters cubed, or cubic centimeters. Now that's the volume of the entire, of the entire fancy, I guess, tea infuser, I call them fancy teabags. But we wanna know what 80% of its capacity is, because that's how much tea we need. So we wanna multiply this times 80%. So you would multiply 0.8 times 1.5 times 7.9 centimeters cubed. And let's see, 0.8 times 1.5. Let's see, that would be .8 plus .4, so this part, this is gonna be 1.2 times 7.9 centimeters cubed. We could try to approximate this. In fact, this is looking approximately right, but we could use a calculate here, so let's just, let's just feel good about it. So if we say 1.2 times 7.9, we get 9.48. 9.48, and if we round to the nearest tenth, well, round to the nearest tenth. We have an eight in the hundreds place, we're gonna round up to 9.5, which is exactly that choice.