Question

Related Rate

Answer 1

we can do this, but it is not really a related rate problem, it is a mix/min problem

Answer 2

is that picture more or less clear? i cannot really draw a better one
the rectangle on the left is supposd to wrap around the circle, which has radius \(r\) making the side length \(2\pi r\) and the height is \(h\)
then the total area of those surfaces (the thing you have to minimize) is
\[S=2\pi rh+\pi r^2\]

Answer 3

oh i see the surface area is given, that makes
\[2\pi r^2h+\pi r^2=12\] and you want to maximize
\[V=\pi r^2 h\]

Answer 4

damn typo, i meant
\[12=2\pi rh+\pi r^2\]

Answer 5

thats about where i got

Answer 6

you can write all of this only in terms of \(r\) by solving
\[12=2\pi rh+\pi r^2\] for \(r\)

Answer 7

damn i am lame tonight, i mean solve it for \(h\)

Answer 8

\[12=2\pi rh+\pi r^2\]\[\frac{12-\pi r^2}{2\pi r}=h\]

Answer 9

got it!

Answer 10

you're doing fine, haha

Answer 11

so plug that into volume?

Answer 12

stick that in to the volume formula, and you will have only one variable
\[V(r)\]

Answer 13

i hate to write it because i am screwing up so much, but before the algebra it would be
\[V(r)=\pi r^2\frac{12-\pi r^2}{2\pi r}\]

Answer 14

ok so then i simplify and take derivative?

Answer 15

cancel and get
\[V(r)=\frac{r(12-\pi r^2)}{2}\]

Answer 16

then set derivative to zero and find max/min?

Answer 17

if i didn't screw that up to
maybe use
\[V(r)=6r-\frac{1}{2}\pi r^3\]

Answer 18

yeah that should do it, but check my algebra, because i think i am braid dead

Answer 19

i think you're correct

Answer 20

imagine!

Answer 21

now i have a question

Answer 22

why the pouty face if you are in hawaii??
jeez ...

Answer 23

hahah its a puppy face!

Answer 24

oooh i see
you good from there right? rest should be routine

Answer 25

i believe so :) thank you

Answer 26

yw