Dimensional Analysis: Basic Conversions-Inches to Centimeters

Now we’re going to do a conversion that crosses unit systems we’re going to go from inches to centimeters imperial system to the metric system we need to know the relationship between centimeters and inches and luckily there is a one-step conversion for that now I don’t require my students to memorize a long list of conversions and I allow them to use a sheet of conversions when you look at those relationships on there and you’ll find the inches to centimeters relationship be very careful about noticing how its presented your relationships between metric prefixes are exact you know a thousand kilometers per meter that’s exact the relationship between units in the imperial system are exact 12 inches per foot three feet per yard when you cross unit systems you do not expect those relationships to be exact this one is exact there is not three significant figures in that it’s an exact relationship will start this problem just like we started the previous problem I’ll put down my given information I’ll put it over 1 so i can clearly separate my numerator and denominator and now i’m going to put the conversion in as a fraction and I let my units guide me so in the given information have units of inches but I want centimeters so when the denominator of the conversion I’m going to put inches so I get inches over inches well I have to put the rest of the conversion in there so when the numerator goes 2.54 centimeters that right most sense set of parentheses that is saying there are 2.54 centimeters in an inch I see that my units of inches canceled so i’m left with centimeters which is what i want and then to get the answer i have to multiply let’s watch the significant figures so the 4.7 two that’s three significant figures ignore the one as we stated that relationship is exact which means my answer should have three significant figures in it now the calculator gives me this well let’s got way more digits I’m allowed so three significant figures one two three 11.9 the digit next to it’s an eight to round that up so you get 11.0 centimeters oh yeah watch the nines the nine rounds up which means 11 becomes 12 so 12.0 centimeters is the right answer and that is actually the standard diameter for a compact disc let me change something what if i had said let’s convert 4.7 20 inches to centimeters you’d set it up and you know what the number well it’s still the same multiplication so it shouldn’t change anything right well look at my answer I get 11.99 centimeters because 4.7 20 has four significant figures in it the centimeters and inches relationship is exact so my answer gets four significant figures so count your digits watch your significant figures all I want to say something about the relationship between inches and centimeters too because that can be a little confusing the u.s. adopted this relationship in 1866 saying that one meter was equivalent to thirty nine point three seven inches well 1959 the US and some other countries signed a treaty agreeing that one yard was going to be exactly equal 2.91 44 meters which is where we get the bottom one now I see the middle of one the inch equals 25 point four millimeters quite often the one that I see in chemistry classes and others for converting is the bottom one but any of those bottom three are equivalent to each other exact relationship but that sets up a problem that one meter is equivalent to thirty nine point three seven inches that’s a u.s. survey inch see when they branched off they actually change the inch two centimeters just by a little bit so the older definition u.s. survey inch definition that if still used in surveying is not 2.54 centimeters it’s a little bit more than 2.54 centimeters per inch so when you’re doing the relationships use the exact 11 inch is exactly equal to 2.54 centimeters well unless you’re a surveyor and Wikipedia actually has a decent summary of the history of an inch

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