Question

For a closed cylinder with radius r cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 8 cm3.

Write the EXACT answers.

r=_____
h=_____

please explain? :) thanks!! @Luigi0210

Answer 1

I use these right?

V= π * r^2 * h

and

A= π * r * (2h+r)

?

Answer 2

and do i set it like this?

8=π * r^2 * h
h=8/π * r^2 ?

Answer 3

Differentiate the SA equation, set it equal to 0 to find where it's a minimum.

Answer 4

it's a closed cylinder, not an opened one. Your surface area formula is incorrect

Answer 5

ahh okay
darn..

so how would we solve this problem then?

Answer 6

Yeah it should be \[\large SA = 2 \pi r h + 2 \pi r^2\]

Answer 7

A = 2pi r^2 + 2pi r h = 2pi r (r + h)

Answer 8

ohhh okay

so then you set it equal to 8?

Answer 9

and V = pi r^2 h

Answer 10

so is it 8=pi r^2 h

so solving for h, h= 8 / pi r^2 ?

Answer 11

yes V = pi r^2 h

Answer 12

8 = pi r^2 h

Answer 13

so solving for h right?

h= 8
--------
π r^2 ?

Answer 14

correct

Answer 15

and then set A' = 0, and solve for r. Once you have r, you can find h.

Answer 16

A' ? the derivative of 2pi r (r + h) ?

Answer 17

not sure what to do here :/

Answer 18

@sourwing ? @agent0smith ?

Answer 19

Plug that equation for h into the SA formula.

Then differentiate w.r.t. r.

Answer 20

wrtr?

Answer 21

so 2πrh+2πr^2

= 2πr(8/π r^2) + 2 π r^2 ?

Answer 22

= 16πr + 2π r^2
-----------------
π r^2 ?

Answer 23

correct. Then find dA/dr

Answer 24

the derivative?

Answer 25

how can I do that? i'm not quite sure :(

Answer 26

@sourwing ?

Answer 27

A = 2πr(8/π r^2) + 2 π r^2
A = 16 / r + 2pi r^2
A' = -16/r^2 + 4pi r

Answer 28

ohhh okay i see

so what do we do from here?

Answer 29

set it equal to 0 and solve for r

Answer 30

okay so

-16
---- + 4πr = 0
r^2

-16/r^2 = -4πr

-16 = -4πr * r^2

-16 = -4πr^3

4/π = r^3 ?

did I do that right so far? @sourwing :/

Answer 31

yes

Answer 32

so r=4/π^(1/3)

Answer 33

no, r = (4/pi)^(1/3)

Answer 34

ahh okay:)

so that's our r value right?

so now we need to find the h?

Answer 35

you had the formula for h, find it, plug r in

Answer 36

so

8
-----
pi ((4/pi)^(1/3))^2 ?

not sure how to simplify that part though :/

Answer 37

@sourwing ?

Answer 38

8
--
pi (4/pi)^(2/3)

Answer 39

oh! and that's as simplified as it gets?

Answer 40

i think so

Answer 41

ahh okay awesome!!Thank you so much!! :D