Question

Find the minimum surface area for a 1 liter cylinder
Use x for the radius

Answer 1

Surface Area = two times the
radius
squared times pi plus two
times the
radius times the height times
pi.
Pi ( ) is approximately
equal to 3.14
Or 22
7
Units of measure are square.

Answer 2

01FormulaReferenceSheet10.pdf

Answer 3

Umm yes I already have all that stuff but I need a definitive way to find the values of the radius and height to get the least possible surface area
I already know all the formulas and stuff

Answer 4

Some sites I've visited suggested finding the derivative but I'm only in Pre-calc and derivatives are in Calculus

Answer 5

Okay do you have a pic of the cylinder or no? Just wondering if not its okay.

Answer 6

No it provides no picture
But it's really simple, it's a cylinder with a radius of x, that is all the information provided

Answer 7

Okay, as long as you have your radius and height, you can go ahead and plug those values into your formula to find your answer for surface area. The formula for surface area gives you the total area of all the surfaces together.

Answer 8

But you see I'm not given a radius or height,
What I am given is the volume, I need to find the radius and height through a method that is preferably not guess and check

Answer 9

Okay just give me a minute and I will show you how. Is that okay?

Answer 10

First of all, what is the volume?

Answer 11

???

Answer 12

I need the volume to continue to help you.

Answer 13

It's 1 liter, I said it in the problem

Answer 14

Okay

Answer 15

Is it multiple choice of type in?

Answer 16

Or neither???

Answer 17

There is no multiple choice
THis is an open ended question,
There are ways to solve this but I prefer not to use guess and check or graphing

Answer 18

@Brill how you get for a cylinder volume equal 1 liter what dimensions need having the cilinder for you get volume equal 1 liter - do you know it ?

Answer 19

Well you can covert it into cubic centimeters, where 1000 cubic centimeters equals 1 liter

Answer 20

Don't know if that's what you wanted

Answer 21

@Brill @jhonyy9 Volume of a cylinder is in other words the capacity of the shape, with the assumption that its walls are extremely thin. It is often expressed in cubic meters (m³) or cubic centimeters (cm³) (1 cm³ = 1 ml). In everyday life it is easier to find the use of liter unit, which is equivalent to 1,000 cm³. A standard sized can used by most of the fizzy drinks manufacturers is in approximation a cylinder with capacity of 330 ml = ⅓ of a liter.

Now when we know what the volume of a cylinder is it would be good to know how to calculate it without the necessity of filling it up with liquid. The calculation turns out to be quite straightforward and requires calculating the area of a circle that forms the base of a cylinder and multiplying the result by the height of that cylinder

Answer 22

Okay, I don't know if any of this helped but if it did just let me know.

Answer 23

Yes I know @briannabug
I already learned that in geometry
But I'm not sure how to calculate the least radius and height

Answer 24

Well not the least radius and height but the least surface area

Answer 25

@Brill so using the above wrote that 1000 cm^3 mean for volume 1 liter you can us it for calculi the dimensions of cylinder for you get like result of volume 1000 cm^3 what mean 1 liter - do you understand it now ?

Answer 26

you know that the volume of a cylinder you get from multiplie area of base time height
- so but you need get it for least radius

area of base what is a circle and area of circle is pi*r^2

sorry are you here ?

Answer 27

Yeah I am sorry
Go on please

Answer 28

Umm sorry what do you mean by
"you can us it for calculi the dimensions of cylinder for you get like result of volume 1000 cm^3 what mean 1 liter"

Answer 29

so 1 liter = 1000 cm^3= 3,14 *r^2 *10
let height of cylinder =10 cm

r^2 = 1000/31,4

r^2 = 31,84

r = sqrt(31,84 = 5,64 cm

radius = 5,64 cm

Answer 30

Ok but why do we make the height equal 10?
Is this not just the same as guess and check?

Answer 31

so for radius 5,64 cm and height of cylinder =10 cm you need to calculi now the total surface area of cylinder - hope you can do it - yes ?

Answer 32

this 10 cm for height is just one example for total surface area - after you calculi this so than you can check it how you can making it smaller than you make smaller what dimensions - ok ?

Answer 33

Yeah that's 554.237263 cm squared

Answer 34

****Find the minimum surface area for a 1 liter cylinder
Use x for the radius
******

the volume of the cylinder is area of the base times its height:
\[ V= \pi x^2h=1 \]
from which we get \( h= \frac{1}{\pi x^2} \)
the total surface area is the area of the top and bottom i.e. 2 pi x^2 plus its side:
\[ A = 2 \pi x^2 + 2 \pi x h \]
or, replacing h:
\[ A= 2 \pi x^2 + \frac{2}{x} \]

that is the surface area of a cylinder as a function of its radius (but with fixed volume=1)
Here is a graph of part of the curve

Answer 35

To find the min value on that curve, take the derivative of A with respect to x and set it equal to zero.

\[ \frac{dA}{dx} = 4 \pi x -2 x^{-2}= 0 \]
solve for x, and use that value to find A

Answer 36

See the problem with using derivatives is that I haven't gotten to them yet in classes so I have no idea what they are,
Is there another way to do this?

Answer 37

Yes, another way is to look at the graph and find the min value, which you can estimate

Answer 38

ok thank you very much!