smallest S.A. given volume 300 cubic cm.

$V_{cylinder}= \pi r ^2 h \\ 300=\pi r^2h\\ h=\frac{300}{\pi r^2}$

so let s put our h=300/(pi r^2)  in $S_{cylinder}=2 \pi rh+2 \pi r^2$

$S_{cylinder}=2 \pi r( \frac{300}{\pi r^2 })+2 \pi r^2\\ S_{cylinder}=\frac{600}{r}+2 \pi r^2 \\ y=\frac{600}{x}+2\pi x^2 \text{ (so the minimum value of x is 3.628) }$

https://www.desmos.com/calculator/mae20kmdjb

so r is 3.628 put it in term s of  $h=\frac{300}{\pi r^2}$

$h=\frac{300}{3.628^2 \pi} \\ h = 7.25499\\ S_{cylinder}=2 \times \pi \times 3.628 \times 7.25499+2\times\pi\times 3.628^2=248.0820699 \dots$

Good work Solveit,

I have not checked it thoroughly but it looks good.

I would suggest that you use calculus to find the minimum rather than using a graph generated by Desmos.

Do you know how to do that?

Here's the Calculus approach.........first, solve the volume in terms of h

300 = pi*r^2 *h

h = [300] /[pi*r^2]  .......now....substitute this into the surface area "formula"

S_area  =  2pi*r^2  +  2pi*r*h

S_area  = 2pi*r^2 + 2pi*r* [300]/[pi*r^2]

S_area = 2pi*r^2  + 600r^(-1)      ....take the derivative with respect to r......

S' _area =  4pi*r - 600r^(-2)    .....set this to 0

4pi*r - 600r^(-2) = 0

4pi*r  = 600r^(-2)

pi*r  = 150r^(-2)

r^3   = 150/pi

r = (150/pi)^(1/3)   = about 3.628     .....as Solveit found .......

And h = 300/ [pi * 3.628^2]  = about 7.255

And the minimum surface area would be just as Solveit found......

Solveit, if you have not done calculus yet your answer was excellent!