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often when you're working with 3d shapes
like a cone you'll be given length
measurements on it and be asked to
calculate the volume of it but sometimes
you're given the volume and you're asked
to work backwards to find what some of
the lengths are and that's what this
problem is doing here it tells you the
volume is one hundred and seventy four
point four it tells you the height is
eight point six meters and asks what is
the radius what would the radius have to
be so that the volume is this and I'd
like to show you two ways to solve this
problem first you have to start with the
volume formula the volume of a cone is
equal to one-third times pi times the
radius squared times the height and
first we'll take these numbers that are
given to us and we'll put them into that
formula so that I know the volume is a
hundred seventy four point four is
that's equal to one-third times pi the
radius
I don't know so I'll just leave that as
R squared times the height of eight
point six and it doesn't say this here
but let's round to nearest tenth so
let's around our answer for radius to
the nearest tenth so at this point
here's I'm going to show you two ways
and the first way has a little bit of a
problem it does introduce some rounding
error because we're gonna round numbers
a couple of times but I found that this
way as long as you're careful this way
seems to help my students they seem to
find it easier and this makes more sense
to my students when they solve it this
first way and if you're careful you can
avoid that rounding error and so here's
what I'm going to do first I know this
volume is equal to one-third times pi
times the radius squared times eight
point six so I got these four terms all
multiplied together R is what I want to
solve for so in this first step let's
simplify this let's I can do this
multiplication in any order let's
multiply these three terms together
first so I have one seventy four point
four is equal to
I'm going to multiply 1/3 times pi times
8.6
when I do I'm going to have 1/3 1/3
times pi I'll use the PI key instead of
typing 3.14 and times 8.6 and I get nine
point zero zero so this one is pretty
easy when rounding to the nearest tenth
but one thing you might want to be
careful of on these is even though what
the final answer is going to be rounded
to the nearest tenth you might want to
round this one to the nearest hundredth
to make sure we don't have too much
rounding error so I'm going to actually
round that to nine point zero one so
this is going to be when I multiply
those three terms together nine point
zero one times R squared so now I want
to solve for R I want to get rid of this
multiplication by nine point zero one so
I'm going to divide this side by nine
point zero one and the benefit of that
is now I have nine point one divided by
nine point one nine point zero one is
just one or leaving R squared by itself
and if I divide this side by nine point
zero one I'm going to divide this side
by nine point zero one and when I do I'm
going to get one hundred and seventy
four point four divided by nine point
zero one and I get nine nineteen point
three five six and again I'm going to
round to the nearest hundredth even
though I only need my final answer to
the nearest tenth so I'm going to say
that's nineteen point three six nineteen
point three six is equal to R squared
but now here's the last part I want to
solve for R so if I know R squared or R
times R is nineteen point three six
what would our have to
to solve for our I'm going to take do
the opposite of squaring I'm going to
take the square root of both sides and
the square root of R squared is just R
and over here I'm going to do the square
root of nineteen point three six and
with my calculator then the square root
of nineteen point three six square root
of nineteen point three six four point
four so this one is going to be four
point four since these measurements
that's in meters and cubic meters this
radius is four point four meters is
going to be the radius of this cone well
it's another way I could do that same
problem one way to do this to avoid that
rounding error that I was talking about
let's try it again so I'm going to
switch a different color here but this
is a hundred and seventy four point four
is equal to one-third times pi times the
radius squared times eight point six so
I'm just rewriting that formula to avoid
that rounding error instead of
multiplying all these things together
and then dividing let's just divide it
right away
let's divide by one-third times pi times
eight point six and when I do I know 1/3
divided by one-third is just one so it
cancels and pi divided by pi is just one
so it cancels an eight point six divided
by eight point six is just one so it
cancels leaving R squared all by itself
and but here's the here's the tricky
part with this one here's the step that
confuses my students sometimes which is
why I often end up having them do it
this way but if I divide this side by
one-third times pi times eight point six
let's divide this side by one-third
times pi times eight point six and but
you have to be careful how you enter
this you have to either do one seventy
four point four divided by one-third
then divided by pi then divided by eight
point six or
group them in parentheses so that the
one you enter into the calculator the
calculator will do all of this
multiplying first then divide that
number into a hundred and seventy four
point four and when you do you get one
seventy four point four divided by I
mean I'll use the parenthesis key 1/3
1/3 times pi times eight point six close
those parentheses equals and again it
looks like I'm getting this time notice
it's nineteen point three six five
that's going to round to nineteen point
three seven if I go to the nearest
hundredth which is a little different
than I had before because of the
rounding that I did up here but when we
round to the nearest tenth though it
shouldn't cause a problem I know if I
take the square root of both sides R
will be equal to the square root of
nineteen point three seven and R will be
equal to the square root of nineteen
point three seven and when I round to
the nearest tenth I've got four point
four zero one one that's to the nearest
tenth it is going to be four point four
meters so that's two ways that you can
solve this problem for the missing
radius one way avoid some of that
rounding error this way seems to make
more sense to my students to kind of
simplify this expression or this
equation before you then divide to get R
by itself so either way I hope you have
good luck with these volume problems
related to a missing radius
