Question

Find the dimensions of a closed right circular cylindrical can of smallest surface area whose volume is 128pi cm^3

Answer 1

I can solve it with calculus.

Answer 2

so the equation you want to minimize is:
\[2\pi r^2 + 2\pi rh \]

you are given the condition that:
\[\pi hr^2 = 128 \pi\]

Answer 3

you can solve the second equation for h by dividing both sides by r^2 and dividing by pi

Answer 4

the equation you want to minimize now has one variable:

\[2\pi r^2 + 256\pi /r\]

Answer 5

the definition of a minimum is where the slope is 0, or where the derivative is 0

Answer 6

you can now take the derivative of this equation. The result is:

\[4\pi r - \frac{256\pi}{r^2} \]

Answer 7

set this equal to 0 and you finally have an equation that you can solve!

Answer 8

factor out 4pi to get:

\[4\pi (r - 64/r^2)\]

Answer 9

Where did you get the 256π/r

Answer 10

find a common denominator:

\[4\pi (\frac{r^3 - 64}{r}) = 0\]

Answer 11

ok

Answer 12

backtrack a little bit

Answer 13

remember the first equation that we solved for h with?

Answer 14

Ok I see. But, how did you get the condition as pi hr^2?

Answer 15

that is the formula for volume, and the volume has to equal 128pi

Answer 16

h is the height and is the radius if it wasn't obvious

Answer 17

wow, yeah i'm a bit tired. So, from the simplified equation I would find the min by setting the derivative equal to zero, and checking that by taking the second derivative, right?

Answer 18

yes

Answer 19

Thank you!

Answer 20

you are welcome