Volume word problems — Basic example

a tea infuser in the shape of a right rectangular pyramid is 7.9 centimeters tall and has a base 3 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 the this tea infuser it's kind of one of those fancy teabags that hold their shape it's in the shape of a right rectangular pyramid so it's gonna look something like this let's see its basis three centimeters by 1.5 centimeters so its basis let's say that's three centimeters by one point five centimeter so that 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 one point five centimeters and then it is seven point nine centimeters tall so if we went from the center and if we were to go straight up it's seven point nine centimeters tall so this dimension right over here is seven point nine seven point nine centimeters and it's a right rectangular pyramid 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 and those but you might have seen these fancy tea infusers so it's gonna look something like this and if it was transparent you would 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 there they say that should be 80% filled with tea so we need to figure out what 80 percent 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 in the middle of the SAT you know not a for proofs but the but the the formula here is in some ways is strangely intuitive you multiply essentially the the the 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 going to 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 you know 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 at times the height and then you divide by three if you didn't divide by three you would get the volume of the cube of the queue or 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 cannot 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 one point five centimeters times height of seven point nine centimeters divided by divided by three well this 3 3 divided by 3 is just 1 and let's see you have centimeters whoops this is let me just change this is centimeters right over here divided by three that cancels with that if centimeters times centimeters times centimeters it's going to be centimeters cubed so your volume is going to be let's see it's going to be one point five times seven point nine 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 tea bags but we want to know what 80 percent of its capacity is it because that's how much tea we need so we want to multiply this times 80 percent so you would multiply zero point eight times one point five times seven point nine centimeters cubed and let's see 0.8 times 1.5 let's see that would be 0.8 plus 0.4 so this is this part this is gonna be one point two times seven point nine centimeters cubed we could try to approximate this in fact this this is looking approximately right but we could use a calculator here so let's just let's just feel good about it so if we say one point two times seven point nine we get nine point four eight nine point four eight and if we round to the nearest tenth well the round to the nearest tenth we have an eight in the hundredths place we're gonna round up to nine point five which is exactly that choice