Question

[25 POINTS] A cylinder has a volume of 400π cm3. Find the dimensions that minimize its surface area.

Answer 1

r = 5.848cm, h = 11.696cm

Explanation:

Surface area:
`SA=2\pi rh+2\pi r^2`

Volume:
`V=\pi r^2h`

Therefore, the height of a cylinder given its volume would be
`h=(V)/(\pi r^2)`, thus:

`SA=2\pi r((V)/(\pi r^2))+2\pi r^2\\\\SA=(2V)/(r)+2\pi r^2\\\\SA=(2(400\pi))/(r)+2\pi r^2\\\\SA=(800\pi)/(r)+2\pi r^2`

We now find the derivative of the surface area of the cylinder with respect to its radius and set it equal to 0, solving for the radius:

`(d(SA))/(dr)=-(800\pi)/(r^2)+4\pi r`

`0=-(800)/(r^2)+4\pi r`

`0=800\pi+4\pi r^3`

`-800\pi=4\pi r^3`

`-200=r^3`

`r\approx5.848`

If
`0If [tex]r>5.848`, then the surface area of the cylinder increases

Therefore, the surface area is minimized when the radius is
`r=5.848cm`, making the minimum height
`h=(V)/(\pi r^2)=(400\pi)/(\pi(5.848)^2)\approx11.696cm`.

In conclusion, the 2nd option is correct.