# Dimensions that minimize the surface area of a cylinder (KristaKingMath)

hi everyone today we’re going to talk about

how to find the dimensions of the cylinder that minimize its surface area to complete

this problem we’ll draw a picture of the problem and write what we know identify optimization

and constraint equations and then use the derivative of the optimization equation to

find the dimensions let’s take a look in this particular optimization problem we’ve been

told that a cylindrical can must be made to hold one liter of oil and then we’ve been

asked to find the dimensions of the can that will minimize the cost of the metal used to

make it as with any optimization problem the first thing you want to do is draw a diagram

of what you’ve been told and write down as much information as you can so we’ve drawn

a diagram of the cylinder and we’ve indicated r as the radius and h as the height of the

cylinder we’ve also written down that one liter is equal to a thousand cubic centimeters

and that the area formula for a cylinder is two pie r h plus two pie r squared we also

know that the volume of a cylinder is pie r squared h so really quickly in case you

don’t remember those remember that the area formula is two pie r which is the circumference

of the circle times h the height of the can so that gives you the outside or the material

used to make the side and then pie r squared is the area of the circle so that will give

you the top and bottom you multiple by two to account for the top and bottom together

so when we simplify we get two pie r h plus two pie r squared the formula for the volume

is the area of the circle times the height of the can so the area of the base times the

height which will give us the total volume of the can once we’ve written down all the

information we’ve been given our next step is to identify a constraint equation and an

optimization equation so with any optimization problem like this you’re always going to need

an equation you can optimize but you’ll also need a constraint equation because we’ve been

asked to minimize the cost of the metal that means that we’re going to be minimizing the

surface area of the can because the metal is defined by the surface area so minimizing

the surface area means that our optimization equation will be the area equation so let’s

go ahead and say that this area equation here will be our optimization equation our constraint

equation is the equation that limits us so we’ve been told that cylindrical can has to

be made to hold one liter of oil that’s the only constraint we’ve been given so what the

problem is asking is for us to optimize the area the surface area minimize the cost of

metal or minimize the cost of the surface area within the constraint or within the condition

that the volume has to hold one liter or a thousand cubic centimeters so this volume

equation here will be our constraint equation our next step is to figure out how to reduce

our optimization equation to just one variable so because we have in our optimization equation

here both r for the radius and h we need to get rid of one of those the way that we’re

going to get rid of one of them is by solving the constraint equation for one of the other

and then plugging in to the optimization equation so let’s go ahead and solve our constraint

equation our value equation for either r squared or h in this case let’s go ahead and solve

it for h because that will be really easy to plug in to the area equation we just got

h right here so the way that we’ll do that is just by plugging one thousand cubic centimeters

in for volume we know that has to be the volume and we’ll set that equal to pie r squared

h and we’ll solve for h obviously to solve for h we’ll just divide both sides by pie

r squared so we’ll get one thousand divided by pie r squared now that we’ve solved for

h we can go ahead and plug that back into our area equation so if we say area over here

is equal to two pie r multiplied by h which we solved for as one thousand over pie r squared

plus two pie r squared now we have our optimization equation for surface area in the form of one

variable r and now that we’ve got it in one variable all we need to do is simplify take

the derivative set the derivative equal to zero and solve for r and r should give us

the radius of the can that minimizes the surface area so let’s go ahead and simplify this first

we’ll see that we get pie to cancel here and we’ll get r to cancel from the numerator and

the squared to cancel in the denominator so we’ll be left just two thousand divided by

r plus two pie r squared let’s go ahead now we need to take the derivative because we’ve

simplified this as much as possible so let’s go ahead and make this a little easier on

ourselves we’ll call this two thousand r to the negative one remember we can move a variable

from the denominator to the numerator just by changing the sign in the exponent from

a positive to a negative so two thousand r to the negative one plus two pie r squared

if you’re not comfortable doing this you can obviously use quotient rule to figure this

out so we’re going to take the derivative a prime and we’ll get negative two thousand

r to the negative two plus four pie r and when we move this r to the negative two back

to the denominator we’ll get negative two thousand and actually let’s go ahead and make

that our second term so we don’t lead with the negative we’ll get four pie r minus two

thousand over r squared so that is our derivative equation now we need to set this equal to

zero and solve for r so if we set it equal to zero let’s at the same time go ahead and

find a common denominator in order to find a common denominator we’re going to be multiplying

this first term by r squared over r squared and we’ll get four pie r cubed we can combine

the fractions minus two thousand all over r squared so now what we can see is that this

right hand side here is going to be equal to zero whenever the numerator is equal to

zero or if the denominator is undefined however if the denominator is undefined that means

r is equal to zero and we know r can’t be equal to zero because we’re talking about

a three dimensional here in real space and if r is equal to zero then the cylinder doesn’t

exist so we can almost just discount that and we’re really just looking at where the

numerator is equal to zero so let’s go ahead and factor out a four and we’ll get four times

pie r cubed minus five hundred obviously now we can see that the right hand side will equal

zero only when pie r cubed minus five hundred is equal to zero so we can really just ignore

the four or divide both sides by four and that will go away and now if we add five hundred

to both sides we’ll get five hundred equals pie r cubed if we divide both sides by pie

we’ll get r cubed equals five hundred divided by pie and if we solve for r by taking the

cube root of both sides we’ll get r equals the cube root of five hundred divided by pie

we know now that this must be the radius that minimizes the surface area of the can if we

want to find the height as well we can just go ahead and plug r into our height equation

it’s not very clean but we’ll get one thousand divided by pie times and now we have r squared

here so we get the cube root of five hundred divided by pie all squared that’s really just

going to give us one thousand divided by pie times five hundred over pie to the two thirds

power so these are the dimensions h here and r here that minimize the surface area and

therefore minimize the cost of the metal used to construct the can so I hope you found that

video helpful if you did like this video down below and subscribe to notified of future

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She's pretty good looking I spose, about a 7 out of 10. Maybe 8 out of ten if it weren't for that lazy eye. Keep up the good work!

god.. I've got a HUGE head ache over this question, all because I did not convert 1L to 1000cm^3

But really I don't think it matters what you use as a unit. Does it?

But even still when I follow your steps to similar problems Im not getting the right answer… probably another editing mistake in the calc book… this really makes me wonder, the teams of people making this calc book must have the combined brain power of a turnip.

thats called A prime or the derivative of the function…in other words, that the calculus part of it.

I….I love you? seriously though very awesome. Thank you so much

You're so welcome! Glad you liked the video. 🙂

Y did u cancel 4 in the end and the r^2 is missing?

i divided both sides by 4. since that gives me 0/4 on the left side, i just have 0. i get 4/4 on the right side, which just leaves me with 1. the r^2 disappears when you take the cube root of both sides. 🙂

Great video!!! Thanks for your help ma'am

you're welcome!! 😀

this… HELPED MEEEEEEEEEEEEEEEEEEE!

I knew how to do the problem, but I didn't know why I did the problem. now I know WHY!

Yay! 🙂

You are amazing!!

awww thanks! 😀

wow that very helpfull….. thanks

Glad I could help! 🙂

thanks again,

You're welcome!

I fully understand now, thank you 😀 Feeling ready for my exam. This was the only chapter I struggled in

I did this but it gave me a larger SA than the original. I feel that I accidentally maximized it but I don't know how I did this.

Thank you. I know u did not forget to test test the value of r , but can u tell me why u did not test it? ^_^

Thank you it really help..

Honestly Calculus is tough. I'm a junior in high school currently in Calculus AB. I would have never thought it would have been this confusing. I was looking for a tutor and you seemed perfect 🙂 , would you consider it?

Thank you, that very helpfull

What is a constraint equation and how do you find it? Btw. great video!

this was soooo helpful!!!

Oh my god thank you so much your video is so great and clear

Thanks. I finally understood my assignment @[email protected]

you are the Best. I don't even know your name but you are the best.

Thank you!!!! You just saved me so much! I was in a time crunch, and your video was concise, easy to understand, and visually appealing. You have my neverending gratitude.

you explain things wonderfully

Hey Krista, how do you know that by solving for r when the equation is set to 0, that will be the absolute min? or is it just understood that that is the absolute min?

very informative and you explained it very well.

Why didn't you take the second derivative to actually check that you had a minimum? The assumption was that you had a minimum, and it's true, but you did not show how to check that. How come? Thank you.

thanks babe just did my homework for me.

Thank you for posting this. Your explanation was very clear and I was able to follow along so I could get my project done.

how could you dislike this video is the only puzzling thing about it… Thanks alot for the post it's always really helpful 😀

Am I the only one who like the video before watching it

you are the best. Thank you so much, seriously.

adorei 😀

preciso de uma professora dessa!

Best video on this topic available on youtube. Many thanks!

That was so helpful! Thanks!

This is fantastic thank you so so much.

very helpful video! now to find the height that would minimize the amount of material for the cylinder or can used would I just plug in the radius I got as the final answer into the height equation??

What about maximum surface area?

This just helped me on my HW! Thank you so much!

Thank you

i need a smart woman like you lol thanks

is this grade 9?

Why must you set the derivative of the SA to 0?

thank u so much,,,But I have a question please ,What if my purpose was to find the Maximum surface?

The exact video I was looking for. Thank you so much for your help! From Fargo, North Dakota (:

So glad I found this video! Gonna ace my midterm this week now!

so cool i love math

Love your videos! They seriously are a huge help…Thank you so much!

Very helpful video! But are the measurements for radius and height in cm or inches? Doing a project that requires a literal construction of the container

i believe that DJ khaled once said about this page and I quote: "You Smart. You Real Smart. Matta fact baby you's a genius." #keystosuccess

Thank you so much for your clear explanation. It helped me alot

I love you!

Muchas Gracias, todo se puede lograr. Amo la Ciencia y la Tecnologia, Siento que todo se puede lograr con tus Clases. Gracias Maestra por su tiempo. ES POSIBLE. Matemáticas un gran misterio.

this video was really helpful thanks Krista

Wow, this is the exact video I needed. Thank you!

What if you were asked to maximize the cost of the metal? how would you do it?

What if you were asked to minimize the Energy of a particle in it's ground state in a cylinder for a given volume?

Thanks ,Now I can sleep 😘

question: I am doing one right now thats the same question except the cylinder has no top. Would I simply subtract (pi)r^2 from the optimization equation or is there more to it then that?

Great vid! Helped me conceptualize the process soooo much better!

hey, i have a doubt, if the derivative gets the maximum value instead of minimum? will we still be able to solve to get minimum value?

I'm struggling to find what my optimization and constraint is. I've been told the cylinder has to have volume V and I need to know the shape of the can that is the most efficient cost and cost is equal to $1/per square cm.

Would my optimization be the minimal cost of the can? Shape of the can is my constraint?

I want to find the lateral surface of a can?

Pretty interesting to know that soda cans are made this way.

I actually clicked like on this video before it started. I'm that confident it'll be good!

I was right!

THANKS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Hi, I am looking for a formula to calculate the lenght of a rope enrolled flat on a deck. Kind of spiral enrollment every boat lover have seen.

I love you and your lecture

APPRECIATE IT LOTS

Thank-you so much from Year 10 Aussie students.You are awesome,Krista!

Great explanation!! It only occurred to me recently that there ought to be a formula for minimizing the surface area of a cylinder, both to reduce construction costs, but also to limit radiation heat loss or gain. Moreover, the height to radius should be ratios, scalable to any size. I love geometry and trig, but never a big fan of calculus. Thanks for a fairly intuitive explanation. I was planning on using a spreadsheet for trial by error or range.

How do you have 2 different colored eyes? That is so cool.

Could someone recommend a video or website where I can learn how she found the derivative, A prime? Thanks!

this was sooo helpfullllll bless you

thank you so much

so how do you find the maximum if that's how you find the minimum dimensions?

Extremely helpful. Thanks.

very easy explanation. Thanks a lot

Really good video. Thanks!

she is pretty and smart

I saw another optimization problem which was asking for the maximization.. in that problem once they found the critical point.. they had to test which interval gives a positive value. So my question is how do u know that answer is the one that minimizes? How come u don’t have to test any interval once u get ur critical point (radius) ??

Would also be interesting to show in the conclusion to the video that area is minimized when h=2r (when the height of the cylinder equals its diameter). Just an interesting observation.

Awesome!

Using your solutions ..The ht is twice the radius …h= 10.8 .,,,,.r= 5.4Will the height always be twice the radius when it asks for dimensions that minimize Surf area of a cylinder?

Mam your way of teaching is quite easy and effective..thank u

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This is so accurate! Thank you so much 😊😊

You smart, but y is it that when i plug my dimension values into the volume equation i ain't getting 1000 cm^3?

a sphere of radius 5 cm is placed on a cylinder of a base of 4 cm find out the volume of part of the square that lie inside the vessel

I was wondering what was the working out between the simplified area and area prime. how was the derivative formed from the equation before?