Question

A tin can (right circular cylinder) with top and bottom is to have volume V . What dimensions (the radius of the bottom and the height) give the minimum total surface
area?

Answer 1

V=h(pi)r^2 , A=h2(pi)r + 2(pi)r^2

Answer 2

solving for h, h=V/((pi)r^2)

Answer 3

\[A=2V/r + 2 \pi r^2\]

Answer 4

Does that look right?

Answer 5

so far, so good.
I assume this is a calculus problem ?

Answer 6

yes

Answer 7

now, I take the derivative of that, and set it equal to zero?

Answer 8

yes

Answer 9

what about still having two variables on RHS?

Answer 10

in this problem, volume is "given" and a fixed value
I would use the notation \(V_0\) to emphasize it is a constant

Answer 11

so, the derivative of V is 0?

Answer 12

if we had to find its derivative, it would be 0
but in your equation, it is a coefficient.

Answer 13

you want to find dA/dr of
\[ \frac{dA}{dr}= \frac{d}{dr}\left(2V_0\ r^{-1} + 2 \pi r^2 \right)\]

Answer 14

\[0 = -2Vr^-2 + 4 \pi r\]

Answer 15

yes

Answer 16

any luck with this ?

Answer 17

sorry, had to step away

Answer 18

came up with \[r=\sqrt[3]{V/2 \pi}\]

Answer 19

which equation do I sub r into to find h