okay today's practice problem is going
to be about dimensional analysis. the first question asks: express one point
zero zero ounces or one ounce in liters, so we are going from ounces to liters
now just to give you the relevant information in order to do this
conversion we will need to know that one liter is equal to one point zero five
six 7 quarts, and we also need to know that one quart is equal to exactly 32
ounces, so that is that is not an approximation that is exactly 32 ounces
just the way that there's exactly one thousand milliliters in a liter
that sort of thing, so that value will not limit the number of significant
figures we are able to use, so bear that in mind that is the relevant information
for that question. and then the next question we want to answer is as follows
a packaging material has a density of forty seven point nine three kilograms
per cubic meter, and we want to express this in grams per cubic centimeter, so we
do have to do some conversions there so if any of this is unclear go ahead and
check out my tutorial on dimensional analysis, it's going to give you all the
strategy that you need to be able to solve these kinds of problems and once
you're ready go ahead and give these a try.
so let's answer this first question
remember we want to convert 1.00 ounces or one ounce into liters, so we have some
conversion factors here that we're going to want to use. now what should we use
first? we want to cancel out ounces right, we don't want ounces we want
liters, so let's use the conversion factor that has ounces in it, that's
going to be one quart equaling thirty-two ounces so let's multiply one
ounce by one quart over 32 ounces. remember we are
always going to be multiplying our value by things that are equal to 1, and we
achieve that by using fractions where the top and the bottom are equivalent to
one another. they're just being expressed differently, because that won't change
our value, it'll just give us different units. now it's very important that we
put ounces on the bottom because that is how we get ounces to cancel. now let's
use the other conversion factor, we have one liter over one point zero five six
seven quarts, and it's very important that we do this this way, so that quarts
will also cancel, so let's go ahead and cancel our units, ounces are gonna cancel
because we have ounces and then we have ounces on the bottom, and then quarts are
also going to cancel because we have quarts on the top and quarts on the
bottom, so just like algebraic variables like X in algebra these units are gonna
cancel this way, and the only unit left is liters on the top and that is what we
wanted. and if we set things up this way and we end up with the units we want we
know we did the problem correctly. so now all we have to do is the arithmetic so
now let's just quickly assess sig figs, we've got three sig figs in the first
value, one point zero zero, those trailing zeros are significant because they are
decimal zeros. the 32 is unlimited as we said that is an exact
conversion, it's thirty two point zero zero zero with infinite zeros so that is
precisely 32 ounces, so that will not limit our sig figs here and then we have
five sig figs in this value so our answer must have three sig figs. those
three sig figs are going to limit our answer to three sig figs. so we go ahead
and plug this into the calculator and we get zero point zero two nine six liters
and that will be our answer. now let's look at the next problem. we have some
packaging material like a Styrofoam of some kind and the density of this
material is forty seven point nine three kilograms per cubic meter. and remember
we want to get this in grams per cubic centimeter so what are we going to do?
let's write down our value, we've got 47.9 three kilograms over
meters cubed and once again we can treat these units just like algebraic
variables, so meters cubed literally means meters times meters times meters
and we can write this out this way which will be useful in helping us wrap our
heads around exactly what to do, and we'll see why in a moment. first let's
take care of kilograms since that's by itself
we don't want kilograms, we want grams so let's use the conversion factor 1,000
grams over one kilogram, because that's going to get rid of kilograms. those will
cancel out and we will end up with grams which is what we want. now let's take
care of the meters, we don't want meters we want centimeters, so let's use the
conversion factor one meter over 100 centimeters because those are equivalent.
there are 100 centimeters in a meter and the reason we do this is because meters
will cancel. however only one of those meters has cancelled, we need to cancel
all three so we will need a second conversion factor of one meter over 100
centimeters that will allow this second meters to cancel, and then we need a
third conversion factor of one meter over 100 centimeters to get the third
meters to cancel, so we needed to multiply by this conversion factor three
times not only to get rid of those three meters comprising the meters cubed but
now we have centimeters times centimeters times centimeters which is
centimeters cubed, which is what we want. so we don't necessarily have to do it
this rigorously but this is just to illustrate the fact that having meters
cubed converting into centimeters cubed is not the same as just meters to
centimeters we cannot use that conversion factor one time we needed to
use it three times or at least one time cubed so that is how we needed to do
this particular problem and now we have grams which is what we wanted and we
have centimeters times centimeters times centimeters we will have an answer in
grams per cubic centimeter which is what we want. and now all we need to do is
plug these numbers into the calculator, we have 47.9 three times a thousand and
then we want to divide by a hundred one two three times. so we want to divide by
a hundred and then divide by hundred and then divide by a hundred again
and if we put all of that in the calculator we will get zero point zero
four seven nine three grams per cubic centimeter. so none of these conversion
factors are gonna change the number of sig figs. we have a thousand we have a
hundred but those are all exact conversions, there's exactly one thousand
grams in a kilogram so we are just going to have the four sig figs that we
started with in the initial value. and so that is why our answer has four sig figs
and that will be the answer to this problem