Question

Write an equation for the following:
A cylindrical can with no top has surface area of 3pi square meters.

This is part of a min/max problem . I just need the 1st part done. I know that the area of an open top cylinder is 2PI* r + PI r^2. so how would you write the equation given that the total area is 3PI square metres

Answer 1

total area includes the variable h (height of the can)

Answer 2

yes- my bad

Answer 3

it should be 2pi*r*h

Answer 4

are you trying to find maximum area of can?

Answer 5

sorry maxim volume

Answer 6

What height h and base radius will maximise the volume of the can?

Answer 7

are you able to walk me through it, I know the method but some help will be great thanks

Answer 8

ill have to leave it in telliots capable hands - i gootta go now

Answer 9

Thanks for your help..

Is the equation - 3PI = 2PI*r*h + PIr^2

Answer 10

Area = 3 pi = pi r^2 + 2 pi r h

Answer 11

okay

Answer 12

Sound right? Now, solve for h

Answer 13

I will 1st have to differentiate right, and then find the roots?

Answer 14

Next is to substitute this value for h into the equation for volume.
Then differentiate and set dV/dr = 0.

Answer 15

okay

Answer 16

and then find the roots right?

Answer 17

What did you get for h?

Answer 18

No roots necessary, the squares will go away when you differentiate.

Answer 19

okay 1 sec

Answer 20

\[h=3\pi-pir^2/2pir\]

Answer 21

The pi's cancel

Answer 22

all 3

Answer 23

Multiply top and bottom by 1/pi. All 3 pi's cancel.

Answer 24

okay, so answer is 3-r^2/2r

Answer 25

do i equate volume formula to 0

Answer 26

Volume = pi r^2 h = pi r^2 times what you got above (3-r^2)/2r

Answer 27

Simplify, then do dV/dr and set that = 0.

Answer 28

yes, i got confused with area for a minute

Answer 29

\[ 3 PI*r^2-PI *r^4/2r\]

Answer 30

So where are we? I made a mistake here the first time through. There are roots.

Answer 31

okay, no problem telliot...
is that what I differentiate?

Answer 32

What I got for the volume is
V = pi r^2 [ (3-r^2) / 2r ]
Does that look right?

Answer 33

yes....do i need to simplify before i differentiate?

Answer 34

Yes. Cancel the r to get
V = pi r (3-r^2) / 2

Answer 35

okay and then multiply the numerator out?

Answer 36

Right. I get 3 pi r / 2 - pi r^3 / 3

Answer 37

sorry, over 2

Answer 38

yes

Answer 39

dV/dr = 3 pi / 2 - 3 pi r^2 / 2 = 0
Sound good?

Answer 40

and then differentiate both terms

Answer 41

Yes

Answer 42

okay

Answer 43

i have

Answer 44

(6pi-3pi*r)/4 - (6pir^2-pir^3)/4

Answer 45

In the end, the answer is extremely simple. You can show it's a max by the 2nd derivative test. And I would check my answer either by graphing or by calculating the volume for your answer plus a little bit and minus a little bit and show they are less than the max.

Answer 46

yes

Answer 47

I should be able to do it from here...just need a bit of time

Answer 48

Thank you so much, you are very helpful

Answer 49

Wait a minute. I had V = 3 pi r / 2 - pi r^3 / 3.
What is dV/dr?

Answer 50

is v=3pir/2 etc what you differentiated?

Answer 51

Yes. We want to know how volume changes as r changes, dV/dr

Answer 52

Gotta run. See my answer for dV/dr previously in the thread. I got finally, r = 1. I would work it all through again from the beginning on a clean sheet of paper. And do the checks I mentioned.

Answer 53

okay thank you, i'll be fine from here, you have been a great help...kind regards

Answer 54

yw

Answer 55

Hi Telliot, I managed to get 1 as well . Thank you very much for your help!

Answer 56

Good. See if you can graph it or calculate volume of 0.99 and 1.01 versus 1.0 for r.