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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