So here in this example question, it says, if the density of an unknown metal is 21.4 grams per centimeter cubed, express its density in pounds per feet cubed. All right. The information they are giving to us is not a given amount. It's actually a conversion factor: 21.4 grams per 1 centimeter cubed, and they are telling us that we need to get to our end amounts, which will be pounds per cubic feet. We can set this up as a dimensional analysis type of question because all we are doing is setting things up with conversion factors, allowing them to cancel out, and wind up with end units or end amounts. All right.
So, we need to find a way of changing grams to pounds and changing centimeters cubed to feet cubed. Let’s do the easier one first. Let's go from grams to pounds. Here we're going to use a new conversion factor. We're already starting out with our first conversion factor. This question actually doesn't have a given amount unit by itself. That can happen. Now we know that there is a connection between grams and pounds. When we talked about the different types of conversion factors, we said here that 1 pound was equal to 453.59 grams. Grams go on the bottom, so that they can cancel out like this. So we've done the easy part. We've converted grams to pounds. Now it's up to us to convert centimeters cubed to feet cubed.
Alright. So, what we're gonna do first is we're gonna say that there is a connection between centimeters and inches. We want to get rid of these centimeters, which are on the bottom, so we actually have to put centimeters here on top. Centimeters and inches are connected, and the relationship is that 1 inch is equal to 2.54 centimeters. However, this is cubed and these centimeters here are not. So you would cube the whole thing. We'll come back and see what effect that has on our numbers. So, basically, centimeters cubed are canceled out, now we have inches cubed. We want to get rid of inches, so we put inches up here. We want feet remember those are connections between inches and feet and that's 1 foot is equal to 12 inches These inches are cubed, but these are not, so I'd have to cube this whole thing. And this would represent my conversion factor. Inches cancel out. Inches cubed cancel out. So what I'd have at the end is pounds over feet cubed, which are the units I'm trying to isolate.
Let’s come down here and see what effect would all of this have. So we’d have 21.4 grams and then we'd have on the bottom 1 centimeter cubed. We'd have 1 pound for 453.59 grams. When we do 2.54 cubed, that's 2.54 times 2.54 times 2.54. That comes out to 16.387 centimeters cubed over 1 cubed is just 1, over inches cubed, and then 12 cubed is 12 times 12 times 12, which is 1728 inches cubed over 1 foot cubed. So conversion factor 1, 2, 3, and 4. Canceling all the units will give us what we need for our end amount, which will be in pounds per feet cubed. If we look, it would be 21.4 multiplied by 16.387 times 1728 divided on the bottom by 453.59. So what we get initially is we would get 1335.97, but remember the number of significant figures in your answer is based on the digits given within the question. 21.4 has within it 3 significant figures, so our answer needs 3 significant figures. To get that, I'd have to convert it to scientific notation, so I go 1, 2, 3 spaces, and this will come out to be 1.34×103 pounds per cubic feet cubic as our final answer. So this would be a way of converting the units of density from one set of values to another set of values. And remember, it's treated like a dimensional analysis question. Use conversion factors in order to isolate your end amounts at the end. It just happens to be here that our end amounts are two units, those of pounds and feet cubed.
